tri lai - math.unl.edutlai3/catalan objects.pdf · catalan numbers c n = 1 n+1 2n n the number...
TRANSCRIPT
![Page 1: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/1.jpg)
Bijection Between Catalan Objects
Tri Lai
University of Nebraska–LincolnLincoln NE, 68588
October 3, 2018
Tri Lai Bijection Between Catalan Objects
![Page 2: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/2.jpg)
Catalan Numbers
Cn =1
n+1
(2nn
)The number of full binary tree with 2n + 1 vertices (i.e., n internalvertices).
The number of triangulations of a convex (n + 2)-gon.
The number of semi-pyramid with n dimers.
Tri Lai Bijection Between Catalan Objects
![Page 3: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/3.jpg)
Catalan Numbers
There are more than 200 such objects!!
Tri Lai Bijection Between Catalan Objects
![Page 4: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/4.jpg)
Three Types of Trees
Binary Trees
Full Binary Trees
Planar Trees
Tri Lai Bijection Between Catalan Objects
![Page 5: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/5.jpg)
Three Types of Trees
Tri Lai Bijection Between Catalan Objects
![Page 6: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/6.jpg)
Three Types of Trees
Tri Lai Bijection Between Catalan Objects
![Page 7: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/7.jpg)
Bijection Between Binary Trees and Full Binary Trees
Theorem
The number of binary trees (not necessarily full) of n vertices is equal tothe number of full binary trees with 2n + 1 vertices.
Tri Lai Bijection Between Catalan Objects
![Page 8: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/8.jpg)
Bijection Between Binary Trees and Full Binary Trees
Theorem
The number of binary trees (not necessarily full) of n vertices is equal tothe number of full binary trees with 2n + 1 vertices.
Hint: We already learned the bijection!
Tri Lai Bijection Between Catalan Objects
![Page 9: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/9.jpg)
Bijection Between Binary Trees and Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 10: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/10.jpg)
Planar Trees
Tri Lai Bijection Between Catalan Objects
![Page 11: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/11.jpg)
The Number of Planar Trees
Theorem
The number of planar trees with n + 1 vertices is Cn.
Exercise: Prove by generating function.
Tri Lai Bijection Between Catalan Objects
![Page 12: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/12.jpg)
The Number of Planar Trees
Theorem
The number of planar trees with n + 1 vertices is Cn.
Exercise: Prove by generating function.
Tri Lai Bijection Between Catalan Objects
![Page 13: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/13.jpg)
The Number of Planar Trees
Theorem
The number of planar trees with n + 1 vertices is Cn.
We need to show
y =∑
n≥0 Cnxn+1 = xf
f is the generating function of the binary tree.
Tri Lai Bijection Between Catalan Objects
![Page 14: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/14.jpg)
The Number of Planar Trees
Theorem
The number of planar trees with n + 1 vertices is Cn.
y = x + xy1−y
xf satisfies the same recurrence.
Tri Lai Bijection Between Catalan Objects
![Page 15: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/15.jpg)
Bijection Between Binary Trees and Planar Trees
Tri Lai Bijection Between Catalan Objects
![Page 16: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/16.jpg)
Bijection Between Binary Trees and Planar Trees
Tri Lai Bijection Between Catalan Objects
![Page 17: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/17.jpg)
Bijection Between Binary Trees and Planar Trees
Tri Lai Bijection Between Catalan Objects
![Page 18: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/18.jpg)
Bijection Between Binary Trees and Planar Trees
Tri Lai Bijection Between Catalan Objects
![Page 19: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/19.jpg)
Three Types of Paths
Dyck Paths
2-Colored Motzkin Paths
Lukasiewicz Paths
Tri Lai Bijection Between Catalan Objects
![Page 20: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/20.jpg)
Dyck Path
A Dyck path of length 2n is a lattice path:
1 From (0, 0) to (2n, 0);
2 Use the element steps ↗ and ↘;
3 Never go below the x-axis.
Tri Lai Bijection Between Catalan Objects
![Page 21: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/21.jpg)
Dyck Path
Theorem
The number of Dyck paths of length 2n is Cn.
Exercise: Prove by generating functions.
Tri Lai Bijection Between Catalan Objects
![Page 22: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/22.jpg)
Dyck Path
Theorem
The number of Dyck paths of length 2n is Cn.
Prove by reflecting principle.
Tri Lai Bijection Between Catalan Objects
![Page 23: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/23.jpg)
Dyck Path
# Dyck paths = # General paths - # ‘Bad’paths
Tri Lai Bijection Between Catalan Objects
![Page 24: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/24.jpg)
Dyck Path
# Dyck paths =(
2nn
)- # ‘Bad’ paths
Tri Lai Bijection Between Catalan Objects
![Page 25: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/25.jpg)
Dyck Path
# Dyck paths =(
2nn
)- # ‘Bad’ paths
Tri Lai Bijection Between Catalan Objects
![Page 26: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/26.jpg)
Dyck Path
# Dyck paths =(
2nn
)- # ‘Bad’ paths
Tri Lai Bijection Between Catalan Objects
![Page 27: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/27.jpg)
Dyck Path
# Dyck paths =(
2nn
)-(
2nn+1
)
Tri Lai Bijection Between Catalan Objects
![Page 28: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/28.jpg)
Motzkin Path
A Motzkin path of length n is a lattice path:
1 From (0, 0) to (n, 0);
2 Use the element steps ↗, ↘, and →;
3 Never go below the x-axis.
Tri Lai Bijection Between Catalan Objects
![Page 29: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/29.jpg)
Motzkin Path
Exercise: Prove the following recurrence for the o.g.f. m of Motzkinpaths:
m = 1 + xm + x2m2
Tri Lai Bijection Between Catalan Objects
![Page 30: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/30.jpg)
2-colored Motzkin Path
A 2-colored Motzkin path of length n is a lattice path:
1 From (0, 0) to (n, 0);
2 Use the element steps ↗, ↘, and →;
3 Never go below the x-axis.
4 The horizontal steps are colored by red or blue.
Theorem
The number of 2-colored Motzkin paths of length n − 1 is Cn.
Exercise: Prove by generating function.
Tri Lai Bijection Between Catalan Objects
![Page 31: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/31.jpg)
Bijection: 2-colored Motzkin Paths and Dyck paths
Tri Lai Bijection Between Catalan Objects
![Page 32: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/32.jpg)
Bijection: 2-colored Motzkin Paths and Dyck paths
Tri Lai Bijection Between Catalan Objects
![Page 33: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/33.jpg)
Lukasiewicz Path
A Lukasiewicz path of length n is a lattice path:
1 From (0, 0) to (n, 0);
2 Use the element steps: ↗ of arbitrary height, ↘ of depth 1, and →;
3 Never go below the x-axis.
Tri Lai Bijection Between Catalan Objects
![Page 34: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/34.jpg)
Lukasiewicz Path
Theorem
The number of Lukasiewicz paths of length n is Cn.
Prove by generating functions.
Tri Lai Bijection Between Catalan Objects
![Page 35: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/35.jpg)
Bijection Between Dyck Paths and Lukasiewicz Paths
Tri Lai Bijection Between Catalan Objects
![Page 36: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/36.jpg)
Bijection Between Dyck Paths and Lukasiewicz Paths
Tri Lai Bijection Between Catalan Objects
![Page 37: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/37.jpg)
Bijection Between Dyck Paths and Lukasiewicz Paths
Tri Lai Bijection Between Catalan Objects
![Page 38: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/38.jpg)
Exercise
Prove bijectively:
1 (4n + 2)Cn = (n + 2)Cn+1
2 (Touchard identity) Cn+1 =∑
1≤i≤bn/2c(n2i
)Ci2
n−2i
Tri Lai Bijection Between Catalan Objects
![Page 39: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/39.jpg)
Bijection Between Trees and Paths
Tri Lai Bijection Between Catalan Objects
![Page 40: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/40.jpg)
Left-first Search (a.k.a. ‘Preoder’)
We visit the trees recursively as follow:
1 The root
2 The left subtree
3 The right subtree
Tri Lai Bijection Between Catalan Objects
![Page 41: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/41.jpg)
Example
1
2
3
45
6 7
8
9
10
11
12
13
14
15 16
17
Tri Lai Bijection Between Catalan Objects
![Page 42: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/42.jpg)
Bijection: Dyck Paths – Full Binary Tree
1
2
3
4
5
6 7
8
9
10
11
12
13
14
15 16
17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Tri Lai Bijection Between Catalan Objects
![Page 43: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/43.jpg)
Bijection: Dyck Paths – Full Binary Tree
1
2
3
4
5
6 7
8
9
10
11
12
13
14
15 16
17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Index the tree by the preoder;
Travel around the tree;
Meet an internal vertex, then go up; meet a leaf, then go down.
Tri Lai Bijection Between Catalan Objects
![Page 44: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/44.jpg)
Bijection: Dyck Paths – Full Binary Tree
1
2
3
4
5
6 7
8
9
10
11
12
13
14
15 16
17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Prove that the map is well-defined.
Tri Lai Bijection Between Catalan Objects
![Page 45: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/45.jpg)
Bijection: Dyck Paths – Full Binary Tree
1
2
3
4
5
6 7
8
9
10
11
12
13
14
15 16
17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Prove that the map is indeed a bijection.
Tri Lai Bijection Between Catalan Objects
![Page 46: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/46.jpg)
Reciprocal Bijection: Dyck Paths – Full Binary Tree
Tri Lai Bijection Between Catalan Objects
![Page 47: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/47.jpg)
Reciprocal Bijection: Dyck Paths – Full Binary Tree
Tri Lai Bijection Between Catalan Objects
![Page 48: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/48.jpg)
Reciprocal Bijection: Dyck Paths – Full Binary Tree
Tri Lai Bijection Between Catalan Objects
![Page 49: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/49.jpg)
Reciprocal Bijection: Dyck Paths – Full Binary Tree
Tri Lai Bijection Between Catalan Objects
![Page 50: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/50.jpg)
Reciprocal Bijection: Dyck Paths – Full Binary Tree
x
Tri Lai Bijection Between Catalan Objects
![Page 51: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/51.jpg)
Reciprocal Bijection: Dyck Paths – Full Binary Tree
x
Tri Lai Bijection Between Catalan Objects
![Page 52: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/52.jpg)
Reciprocal Bijection: Dyck Paths – Full Binary Tree
x
x x
x
Tri Lai Bijection Between Catalan Objects
![Page 53: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/53.jpg)
Reciprocal Bijection: Dyck Paths – Full Binary Tree
*
x
x x
x x
x
x x
Tri Lai Bijection Between Catalan Objects
![Page 54: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/54.jpg)
Bijection: Binary Trees and Motzkin Paths
1
2
3
45
6 7
910
11
1213
14
8
Tri Lai Bijection Between Catalan Objects
![Page 55: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/55.jpg)
Bijection: Binary Trees and Motzkin Paths
1
2
3
45
6 7
910
11
1213
14
8
Tri Lai Bijection Between Catalan Objects
![Page 56: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/56.jpg)
Bijection: Binary Trees and Motzkin Paths
1
2
3
45
6 7
910
11
1213
14
8
1 2 3 4 5 6 7 8 9 10 11 12 13
Exercise:Prove that the map is well-defined and that the map is bijective.
Tri Lai Bijection Between Catalan Objects
![Page 57: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/57.jpg)
Bijection: Planar Trees and Lukasiewicz Paths
14
1
2
3
4
5
6
7
8
910
11
12
13
14 15
1 2 3 4 5 6 7 8 9 10 11 12 13
The height of step i is d(i)− 1.
Tri Lai Bijection Between Catalan Objects
![Page 58: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/58.jpg)
Bijection: Planar Trees and Dyck Paths
Tri Lai Bijection Between Catalan Objects
![Page 59: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/59.jpg)
Bijection: Planar Trees and Dyck Paths
Tri Lai Bijection Between Catalan Objects
![Page 60: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/60.jpg)
Bijection: Planar Trees and Dyck Paths
Tri Lai Bijection Between Catalan Objects
![Page 61: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/61.jpg)
Bijection: Planar Trees and Dyck Paths
Tri Lai Bijection Between Catalan Objects
![Page 62: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/62.jpg)
Bijection: Planar Trees and Dyck Paths
Tri Lai Bijection Between Catalan Objects
![Page 63: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/63.jpg)
Bijection on Staircase Polygons
Tri Lai Bijection Between Catalan Objects
![Page 64: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/64.jpg)
The Number of Staircase Polygons
Theorem
The number of staircase polygons of perimeters 2n + 2 is equal to Cn.
Tri Lai Bijection Between Catalan Objects
![Page 65: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/65.jpg)
Bijection: Staircase Polygons – 2-colored Motzkin Paths
Tri Lai Bijection Between Catalan Objects
![Page 66: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/66.jpg)
Bijection: Staircase Polygons – 2-colored Motzkin Paths
Tri Lai Bijection Between Catalan Objects
![Page 67: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/67.jpg)
Bijection: Staircase Polygons – 2-colored Motzkin Paths
Tri Lai Bijection Between Catalan Objects
![Page 68: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/68.jpg)
Bijection: Staircase Polygons – Dyck Paths
2
Tri Lai Bijection Between Catalan Objects
![Page 69: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/69.jpg)
Bijection: Staircase Polygons – Dyck Paths
3 32
Tri Lai Bijection Between Catalan Objects
![Page 70: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/70.jpg)
Bijection: Staircase Polygons – Dyck Paths
4
23 3
4
Tri Lai Bijection Between Catalan Objects
![Page 71: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/71.jpg)
Bijection: Staircase Polygons – Dyck Paths
1
1
31
2 13
2
Tri Lai Bijection Between Catalan Objects
![Page 72: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/72.jpg)
Bijection: Staircase Polygons – Dyck Paths
1
13
21
1
3
Tri Lai Bijection Between Catalan Objects
![Page 73: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/73.jpg)
Bijection: Staircase Polygons – Dyck Paths
21
31
3 1
Tri Lai Bijection Between Catalan Objects
![Page 74: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/74.jpg)
Bijection: Staircase Polygons – Dyck Paths
21
31
1
Tri Lai Bijection Between Catalan Objects
![Page 75: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/75.jpg)
Bijection: Staircase Polygons – Dyck Paths
21
31
Tri Lai Bijection Between Catalan Objects
![Page 76: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/76.jpg)
Non-crossing partitions
A non-crossing partition of {1, 2, . . . , n} is a set partition{B1,B2, . . . ,Bk} such that: There are no a < b < c < d with a, c ∈ Bi
and b, d ∈ Bj (i 6= j).
Tri Lai Bijection Between Catalan Objects
![Page 77: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/77.jpg)
Non-crossing partitions
Visualize the n-set S as the vertex set of a regular n-gon.
Each subset (or block) in a set partition of S is a polygon containingthe corresponding vertices.
A non-crossing partition is a set partitions such that the polygonscorresponding to the blocks are non-intersecting.
Tri Lai Bijection Between Catalan Objects
![Page 78: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/78.jpg)
Non-crossing partitions
Tri Lai Bijection Between Catalan Objects
![Page 79: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/79.jpg)
Non-crossing partitions
Tri Lai Bijection Between Catalan Objects
![Page 80: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/80.jpg)
Bijection: Non-crossing partitions– Dyck Paths
12
11
10
9
87
6
5
4
32
1
13
Tri Lai Bijection Between Catalan Objects
![Page 81: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/81.jpg)
Bijection: Non-crossing partitions– Dyck Paths
13
4 12
11
10
9
6
5
7
32
1
13
12
11
10
9
87
6
5
4
32
1
8
Tri Lai Bijection Between Catalan Objects
![Page 82: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/82.jpg)
Bijection: Non-crossing partitions– Dyck Paths
{5,6,8,9}{7}{1,3,4}{2}
8
13
4 1211
10
9
6
5
7
32
1
{10,11,12,13}
Tri Lai Bijection Between Catalan Objects
![Page 83: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/83.jpg)
Bijection: Non-crossing partitions– Dyck Paths
13
12
11
10
9
8 6
5
4
3
2
1
{10,11,12,13}{5,6,8,9}{7}{1,3,4}{2}
8
13
4 1211
10
9
6
5
7
32
1
��������������������������������������������
��������������������������������������������
��������������������������������
��������������������������������
7
Homework: Prove that this map is well-defined and is a bijection.
Tri Lai Bijection Between Catalan Objects
![Page 84: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/84.jpg)
Chord Diagrams
12
3
4
5
6
7
8
9
10
11
121314
15
16
17
18
19
20
21
22
23
24
2526
Paring 2n vertices around the circle by chords;
The chords are non-intersecting.
Tri Lai Bijection Between Catalan Objects
![Page 85: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/85.jpg)
Chord Diagrams
12
3
4
5
6
7
8
9
10
11
121314
15
16
17
18
19
20
21
22
23
24
2526
Theorem
The number of chord diagrams of 2n vertices is Cn.
Tri Lai Bijection Between Catalan Objects
![Page 86: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/86.jpg)
Bijection: Dyck Paths – Chord Diagrams
24
12 3 4
5 678 9
1011 12
13
14 15161718
19
20
22 23
2526
21
Tri Lai Bijection Between Catalan Objects
![Page 87: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/87.jpg)
Bijection: Dyck Paths – Chord Diagrams
1415
16
17
18
19
20
21
22
23
24
2526
12 3 4
5 678 9
1011 12
13
14 15161718
19
20
22 23
2526
21 24
12
3
4
5
6
7
8
9
10
11
1213
Tri Lai Bijection Between Catalan Objects
![Page 88: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/88.jpg)
Parenthesis System
Legal: ((()())())(()()), ((())()(()()))()()Ilegal: (()(())))()()((), ((()(()()))))(()()(())
Theorem
The number of (legal) systems of n pairs of parentheses is Cn.
Tri Lai Bijection Between Catalan Objects
![Page 89: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/89.jpg)
Bijection: Dyck Paths –Parenthesis System
Legal: ((()())())(()()), ((())()(()()))()()Ilegal: (()(())))()()((), ((()(()()))))(()()(())
)
12 3 4
5 678 9
1011 12
13
14 15161718
19
20
22 23
2526
21 24
( ( ) ( ( ) ) ) ( ( ( ) ( ( ) ) ) ) ( ( ( ( ) ) )
Tri Lai Bijection Between Catalan Objects
![Page 90: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/90.jpg)
Triangulation of convex polygon
Theorem
The number of triangulations of a convex (n + 2)-gon is Cn.
Tri Lai Bijection Between Catalan Objects
![Page 91: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/91.jpg)
Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 92: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/92.jpg)
Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 93: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/93.jpg)
Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 94: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/94.jpg)
Reciprocal Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 95: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/95.jpg)
Reciprocal Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 96: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/96.jpg)
Reciprocal Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 97: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/97.jpg)
Reciprocal Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 98: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/98.jpg)
Reciprocal Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 99: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/99.jpg)
Reciprocal Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 100: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/100.jpg)
Reciprocal Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 101: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/101.jpg)
Reciprocal Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 102: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/102.jpg)
Reciprocal Bijection: Triangulations – Full Binary Trees
Tri Lai Bijection Between Catalan Objects
![Page 103: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/103.jpg)
Ordered Pair of Increasing Sequences
A pair of increasing sequences 0 < a1 < a2 < . . . < ak < n and0 < b1 < b2 < . . . < bk < n is said to be ordered if ai ≥ bi .Example:
2 4 7 12 14 171 2 7 11 12 13
Theorem
The number of ordered pairs of sequence of order n is Cn.
Tri Lai Bijection Between Catalan Objects
![Page 104: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/104.jpg)
Bijection: Ordered Pairs –Dyck Paths
20
20 4 7 12 14 17 20
12
111213
7
Tri Lai Bijection Between Catalan Objects
![Page 105: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/105.jpg)
Bijection: Ordered Pairs –Dyck Paths
20
20
7
131211
2
20 4 7 12 14 17
1
Tri Lai Bijection Between Catalan Objects
![Page 106: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/106.jpg)
Exercise
Prove bijectively:
1 (4n + 2)Cn = (n + 2)Cn+1
2 (Touchard identity) Cn+1 =∑
1≤i≤bn/2c(n2i
)Ci2
n−2i
Tri Lai Bijection Between Catalan Objects
![Page 107: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/107.jpg)
Prove (4n + 2)Cn = (n + 2)Cn+1
Pick a vertex randomly from a full binary tree with 2n + 1 vertices.
Tri Lai Bijection Between Catalan Objects
![Page 108: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/108.jpg)
Prove (4n + 2)Cn = (n + 2)Cn+1
Choose one of two options: slide the subtree to the left or to the right.
Tri Lai Bijection Between Catalan Objects
![Page 109: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/109.jpg)
Prove (4n + 2)Cn = (n + 2)Cn+1
Add the missing leaf, and mark it.
Tri Lai Bijection Between Catalan Objects
![Page 110: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/110.jpg)
Prove (4n + 2)Cn = (n + 2)Cn+1
We get a full binary tree with 2n + 3 vertices and a marked leaf.
Tri Lai Bijection Between Catalan Objects
![Page 111: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/111.jpg)
Prove (4n + 2)Cn = (n + 2)Cn+1: Second solution.
3
n+2
n+3 12
Pick a base in a (n + 3)-gon.Tri Lai Bijection Between Catalan Objects
![Page 112: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/112.jpg)
Prove (4n + 2)Cn = (n + 2)Cn+1: Second solution.
2
3
n+2
n+3 1
Pick a base in a (n + 3)-gon, then mark randomly an edge of the polygondifferent from the base.Tri Lai Bijection Between Catalan Objects
![Page 113: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/113.jpg)
Prove (4n + 2)Cn = (n + 2)Cn+1: Second solution.
3
n+2
n+1n+1
1
2
3
n+2
n+31
2
Collapse the marked edge to obtain a triangulation of a (n + 2)-gon,marked and orient the merged edge or diagonal.
Tri Lai Bijection Between Catalan Objects
![Page 114: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/114.jpg)
Exercise
Prove bijectively:
1 (4n + 2)Cn = (n + 2)Cn+1
2 (Touchard identity) Cn+1 =∑
1≤i≤bn/2c(n2i
)Ci2
n−2i
Tri Lai Bijection Between Catalan Objects
![Page 115: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/115.jpg)
Prove Touchard Identity
Work on the RHS:
Pick a Dyck path of length 2i in Ci ways.
Tri Lai Bijection Between Catalan Objects
![Page 116: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/116.jpg)
Prove Touchard Identity
Pick a 2i-subset of {1, 2, . . . , n} in(n2i
)ways.
Tri Lai Bijection Between Catalan Objects
![Page 117: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/117.jpg)
Prove Touchard Identity
Pick a 2i-subset of {1, 2, . . . , n} in(n2i
)ways.
Tri Lai Bijection Between Catalan Objects
![Page 118: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/118.jpg)
Prove Touchard Identity
Work on the RHS:
Break the Dyck path.
Tri Lai Bijection Between Catalan Objects
![Page 119: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/119.jpg)
Prove Touchard Identity
Work on the RHS:
Add n − 2i horizontal steps.
Tri Lai Bijection Between Catalan Objects
![Page 120: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/120.jpg)
Prove Touchard Identity
Work on the RHS:
Color n − 2i horizontal steps in 2n−2i ways.
Tri Lai Bijection Between Catalan Objects
![Page 121: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/121.jpg)
Prove Touchard Identity
Work on the RHS:
Pick a Dyck path of length 2i in Ci ways.
Pick a 2i-subset of {1, 2, . . . , n} in(n2i
)ways.
Break the Dyck path and add n − 2i horizontal steps.
Color n − 2i horizontal steps in 2n−2i ways.
Get a 2-colored Motzkin paths of length n.
Corollary
The number of 2-colored Motzkin paths of length n with n− 2i flat stepsis(n2i
)Ci2
n−2i .
Tri Lai Bijection Between Catalan Objects
![Page 122: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/122.jpg)
Restricted Dyck paths
Theorem
The number of UUU-free Dyck paths of length 2n is Mn, the number of(monochromatic) Motzkin paths of length n.
Exercise: Prove the theorem.
Tri Lai Bijection Between Catalan Objects
![Page 123: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/123.jpg)
Restricted Dyck paths
Theorem
(i) The number of Dyck paths of length 2n containing exactly k ‘UDU’sis(n−1k
)Mn−1−k .
(ii) The number of Dyck paths of length 2n containing exactly k ‘DDU’sis(n−12k
)2n−1−2kCk .
Tri Lai Bijection Between Catalan Objects
![Page 124: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/124.jpg)
Bijection: 2-colored Motzkin paths – Restricted Dyckpaths (Callan 2004)
Append a D step. Replace D by UDD.
Tri Lai Bijection Between Catalan Objects
![Page 125: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/125.jpg)
Bijection: 2-colored Motzkin paths – Restricted Dyck paths
Replace F by UD.
Tri Lai Bijection Between Catalan Objects
![Page 126: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/126.jpg)
Bijection: 2-colored Motzkin paths – Restricted Dyck paths
Replace F by U and insert a D immediately before its associated downstep. Remove the last D step.
Tri Lai Bijection Between Catalan Objects
![Page 127: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/127.jpg)
Bijection: 2-colored Motzkin paths – Restricted Dyck paths
Exercise:
1 Prove that this map is indeed a bijection.
2 # flat steps in 2-colored Motzkin path = # UDUs in the Dyck path.
3 # down steps in 2-colored Motzkin path = # DDUs in the Dyckpath.
Tri Lai Bijection Between Catalan Objects
![Page 128: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/128.jpg)
Proof of part (i)
# Dyck paths of length 2n with k UDUs= # 2-colored Motzkinpaths of length n − 1 with k flat steps.
Cut off these k flat steps from the Motzkin path, we get amonochromatic Motzkin path of length n − 1− k.
# 2-colored Motzkin paths of length n − 1 with k flat steps=(n−1k
)×# monochromatic Motzkin path of length n − 1− k.
# Dyck paths of length 2n with k UDUs=(n−1k
)×#
monochromatic Motzkin path of length n − 1− k
Tri Lai Bijection Between Catalan Objects
![Page 129: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/129.jpg)
Proof of part (i)
# Dyck paths of length 2n with k UDUs= # 2-colored Motzkinpaths of length n − 1 with k flat steps.
Cut off these k flat steps from the Motzkin path, we get amonochromatic Motzkin path of length n − 1− k.
# 2-colored Motzkin paths of length n − 1 with k flat steps=(n−1k
)×# monochromatic Motzkin path of length n − 1− k.
# Dyck paths of length 2n with k UDUs=(n−1k
)×#
monochromatic Motzkin path of length n − 1− k
Tri Lai Bijection Between Catalan Objects
![Page 130: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/130.jpg)
Proof of part (i)
# Dyck paths of length 2n with k UDUs= # 2-colored Motzkinpaths of length n − 1 with k flat steps.
Cut off these k flat steps from the Motzkin path, we get amonochromatic Motzkin path of length n − 1− k.
# 2-colored Motzkin paths of length n − 1 with k flat steps=(n−1k
)×# monochromatic Motzkin path of length n − 1− k.
# Dyck paths of length 2n with k UDUs=(n−1k
)×#
monochromatic Motzkin path of length n − 1− k
Tri Lai Bijection Between Catalan Objects
![Page 131: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/131.jpg)
Proof of part (i)
# Dyck paths of length 2n with k UDUs= # 2-colored Motzkinpaths of length n − 1 with k flat steps.
Cut off these k flat steps from the Motzkin path, we get amonochromatic Motzkin path of length n − 1− k.
# 2-colored Motzkin paths of length n − 1 with k flat steps=(n−1k
)×# monochromatic Motzkin path of length n − 1− k.
# Dyck paths of length 2n with k UDUs=(n−1k
)×#
monochromatic Motzkin path of length n − 1− k
Tri Lai Bijection Between Catalan Objects
![Page 132: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/132.jpg)
Proof of part (ii)
# Dyck paths of length 2n with k DDUs = # 2-colored Motzkinpaths of length n − 1 with k down steps.
(Corollary in proof of Touchard Identity) # 2-colored Motzkin pathsof length n − 1 with k down steps=
(n−12k
)2n−1−2kCk .
# Dyck paths of length 2n with k DDUs =(n−12k
)2n−1−2kCk .
Tri Lai Bijection Between Catalan Objects
![Page 133: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/133.jpg)
Proof of part (ii)
# Dyck paths of length 2n with k DDUs = # 2-colored Motzkinpaths of length n − 1 with k down steps.
(Corollary in proof of Touchard Identity) # 2-colored Motzkin pathsof length n − 1 with k down steps=
(n−12k
)2n−1−2kCk .
# Dyck paths of length 2n with k DDUs =(n−12k
)2n−1−2kCk .
Tri Lai Bijection Between Catalan Objects
![Page 134: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/134.jpg)
Proof of part (ii)
# Dyck paths of length 2n with k DDUs = # 2-colored Motzkinpaths of length n − 1 with k down steps.
(Corollary in proof of Touchard Identity) # 2-colored Motzkin pathsof length n − 1 with k down steps=
(n−12k
)2n−1−2kCk .
# Dyck paths of length 2n with k DDUs =(n−12k
)2n−1−2kCk .
Tri Lai Bijection Between Catalan Objects
![Page 135: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/135.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 136: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/136.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 137: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/137.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 138: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/138.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 139: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/139.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 140: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/140.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 141: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/141.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 142: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/142.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 143: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/143.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 144: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/144.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 145: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/145.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 146: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/146.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 147: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/147.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 148: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/148.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 149: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/149.jpg)
Bijection: Dyck paths – Semi-pyramids
Tri Lai Bijection Between Catalan Objects
![Page 150: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/150.jpg)
Bijection: Dyck paths – Semi-pyramids
Exercise: Find the reciprocal bijection.
Tri Lai Bijection Between Catalan Objects
![Page 151: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/151.jpg)
The height of Dyck paths
The height of a Dyck path is its maximum level.
Tri Lai Bijection Between Catalan Objects
![Page 152: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/152.jpg)
The logarithmic height of Dyck paths
A Dyck path w has the height h(w). Then the logarithmic height `h(w)is
blog2(1 + h(w))c
Tri Lai Bijection Between Catalan Objects
![Page 153: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/153.jpg)
The logarithmic height of Dyck paths
`h(w) = k
⇔ 2k − 1 ≤ h(w)) ≤ 2k+1 − 1
Tri Lai Bijection Between Catalan Objects
![Page 154: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/154.jpg)
The logarithmic height of Dyck paths
Theorem
The number of Dyck paths of length 2n with logarithmic height k = Thenumber of full binary trees on n internal vertices and with Strahlernumber k.
Tri Lai Bijection Between Catalan Objects
![Page 155: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/155.jpg)
Strahler number
k’k
k+1
kk
3
2
2
2
2
2
2
2
1
11
1
11
11
1
max(k, k’)
Tri Lai Bijection Between Catalan Objects
![Page 156: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/156.jpg)
Strahler number
k’k
k+1
kk
3
2
2
2
2
2
2
2
1
11
1
11
11
1
max(k, k’)
S(t, x) =∑
n,k Sn,kxktn
Tri Lai Bijection Between Catalan Objects
![Page 157: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/157.jpg)
Strahler number
k’k
k+1
kk
3
2
2
2
2
2
2
2
1
11
1
11
11
1
max(k, k’)
Frangon (1984)Knuth (2005)
S(t, x) = 1 + xt1−2tS
((t
1−2t
)2, x)
Tri Lai Bijection Between Catalan Objects
![Page 158: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/158.jpg)
Bijection increasing the Strahler number
Tri Lai Bijection Between Catalan Objects
![Page 159: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/159.jpg)
Bijection increasing the Strahler number
Replace each vertex by a zigzag line.
Tri Lai Bijection Between Catalan Objects
![Page 160: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/160.jpg)
Bijection increasing the Strahler number
Replace each vertex by a zigzag line.
Tri Lai Bijection Between Catalan Objects
![Page 161: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/161.jpg)
Bijection increasing the Strahler number
Tri Lai Bijection Between Catalan Objects
![Page 162: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/162.jpg)
Bijection increasing the Strahler number
Branching out along the zigzag lines.
Tri Lai Bijection Between Catalan Objects
![Page 163: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/163.jpg)
Dyck paths
We can do the same with Dyck paths
Tri Lai Bijection Between Catalan Objects
![Page 164: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/164.jpg)
Bijection increasing the logarithmic height
Replace each vertex by a 2-colored horizontal path.
Tri Lai Bijection Between Catalan Objects
![Page 165: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/165.jpg)
Bijection increasing the logarithmic height
Replace each vertex by a 2-colored horizontal path.
Tri Lai Bijection Between Catalan Objects
![Page 166: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/166.jpg)
Bijection increasing the logarithmic height
Replace each vertex by a 2-colored horizontal path.
Tri Lai Bijection Between Catalan Objects
![Page 167: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/167.jpg)
Bijection increasing the logarithmic height
Obtaining a 2-colored Motzkin path.
Tri Lai Bijection Between Catalan Objects
![Page 168: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/168.jpg)
Bijection increasing the logarithmic height
Converting back a Dyck path.
Tri Lai Bijection Between Catalan Objects
![Page 169: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/169.jpg)
Bijection increasing the logarithmic height
Converting back a Dyck path.Exercise: Prove that the process increases the logarithmic height 1 unit.
Tri Lai Bijection Between Catalan Objects
![Page 170: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/170.jpg)
(Another) Chord diagram
Tri Lai Bijection Between Catalan Objects
![Page 171: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/171.jpg)
Multiplying Two Chord Diagrams
x = ?
Tri Lai Bijection Between Catalan Objects
![Page 172: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/172.jpg)
Multiplying Two Chord Diagrams
Tri Lai Bijection Between Catalan Objects
![Page 173: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/173.jpg)
Multiplying Two Chord Diagrams
Tri Lai Bijection Between Catalan Objects
![Page 174: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/174.jpg)
Multiplying Two Chord Diagrams
=
Tri Lai Bijection Between Catalan Objects
![Page 175: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/175.jpg)
Multiplying Two Chord Diagrams
x = ?
Tri Lai Bijection Between Catalan Objects
![Page 176: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/176.jpg)
Multiplying Two Chord Diagrams
Tri Lai Bijection Between Catalan Objects
![Page 177: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/177.jpg)
Multiplying Two Chord Diagrams
= q x
Tri Lai Bijection Between Catalan Objects
![Page 178: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/178.jpg)
Temperley - Lieb Algebra
The Temperley-Lieb algebra TLn(q) generated by {e1, e2, . . . , en−1}:1 e2i = qe i2 e ie j = e je i if |i − j | > 1
3 e ie i+1e i = e ie i−1e i = e i
Tri Lai Bijection Between Catalan Objects
![Page 179: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/179.jpg)
Diagram of e i
i+1
ie
ni21
=
Tri Lai Bijection Between Catalan Objects
![Page 180: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/180.jpg)
Diagram of e i
eq2
ei
=i
Tri Lai Bijection Between Catalan Objects
![Page 181: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/181.jpg)
Diagram of e i
ji+1i
|i-j|>1
j eeij
j j+1i+1i
=
ee i =
j+1
Tri Lai Bijection Between Catalan Objects
![Page 182: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/182.jpg)
Diagram of e i
i+1
ii+1
i+1
eei
i+2i
=
eei
=
i
Tri Lai Bijection Between Catalan Objects
![Page 183: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/183.jpg)
Exercise
Simplify: e1e2e4e2e1e3e2e3e2e4
Tri Lai Bijection Between Catalan Objects
![Page 184: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/184.jpg)
Exercise
1 2 3 4 5
e e e e e e e e1 2 4 2 1 3 2 3 2 4
e e
Tri Lai Bijection Between Catalan Objects
![Page 185: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/185.jpg)
Exercise
1 2 3 4 5
e e e e e e e e1 2 4 2 1 3 2 3 2 4
e e e e e1 2 4
= q
Tri Lai Bijection Between Catalan Objects
![Page 186: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/186.jpg)
Temperley -Lieb Algebra and Heap
q3 2 4e e e e e1 2 4=
1 2 3 4 5
e e e e e e e e1 2 4 2 1 3 2
Tri Lai Bijection Between Catalan Objects
![Page 187: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/187.jpg)
Temperley -Lieb Algebra and Heap
The Normalized Temperley-Lieb algebra NTLn(q) generated by{e1, e2, . . . , en−1} is a Temperley-Lieb algebra with the following patternsforbidden
1 e2i2 e ie i+1e i3 e ie i−1e i
Tri Lai Bijection Between Catalan Objects
![Page 188: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/188.jpg)
Temperley -Lieb Algebra and Heap
q3 2 4e e e e e1 2 4=
1 2 3 4 5
e e e e e e e e1 2 4 2 1 3 2
Tri Lai Bijection Between Catalan Objects
![Page 189: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/189.jpg)
Temperley -Lieb Algebra and Heap
Tri Lai Bijection Between Catalan Objects
![Page 190: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/190.jpg)
Temperley -Lieb Algebra and Heap
Tri Lai Bijection Between Catalan Objects
![Page 191: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/191.jpg)
Temperley -Lieb Algebra and Heap
Tri Lai Bijection Between Catalan Objects
![Page 192: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/192.jpg)
Temperley -Lieb Algebra and Heap
x= q2
Tri Lai Bijection Between Catalan Objects
![Page 193: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/193.jpg)
Temperley -Lieb Algebra and Heap
x= q2
Tri Lai Bijection Between Catalan Objects
![Page 194: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/194.jpg)
Staircase Decomposition
Tri Lai Bijection Between Catalan Objects
![Page 195: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/195.jpg)
Staircase Decomposition
Tri Lai Bijection Between Catalan Objects
![Page 196: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/196.jpg)
Staircase Decomposition (Strict Heap)
Tri Lai Bijection Between Catalan Objects
![Page 197: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/197.jpg)
Staircase Decomposition (Strict Heap)
Tri Lai Bijection Between Catalan Objects
![Page 198: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/198.jpg)
Staircase Decomposition (Strict Heap)
e1
e e
e
e e
e
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 4
6
7 10
14
e
e
e
e e
e
e2
3
4
6 11
12
15
Tri Lai Bijection Between Catalan Objects
![Page 199: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/199.jpg)
Staircase Decomposition (Strict Heap)
e1
e e
e
e e
e
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 4
6
7 10
14
e
e
e
e e
e
e2
3
4
6 11
12
15
Lemma
The the higher staircase has the top (bottom) dimer strictly to the left ofthat of the lower staircase.
Tri Lai Bijection Between Catalan Objects
![Page 200: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/200.jpg)
Staircase Decomposition (Strict Heap)
e1
e e
e
e e
e
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2 4
6
7 10
14
e
e
e
e e
e
e2
3
4
6 11
12
15
1 2 4 6 7 10 142 3 4 6 11 12 15
Tri Lai Bijection Between Catalan Objects
![Page 201: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/201.jpg)
Recall: Bijection: Ordered Pairs –Dyck Paths
20
20
7
131211
2
20 4 7 12 14 17
1
Tri Lai Bijection Between Catalan Objects
![Page 202: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/202.jpg)
Strict Heaps – Staircase Polygons
Tri Lai Bijection Between Catalan Objects
![Page 203: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/203.jpg)
Strict Heaps – Staircase Polygons
Tri Lai Bijection Between Catalan Objects
![Page 204: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/204.jpg)
Strict Heaps – Staircase Polygons
Tri Lai Bijection Between Catalan Objects
![Page 205: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/205.jpg)
Strict Heaps – Staircase Polygons
Tri Lai Bijection Between Catalan Objects
![Page 206: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/206.jpg)
Strict Heaps – Staircase Polygons
Tri Lai Bijection Between Catalan Objects
![Page 207: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/207.jpg)
Heaps – Permutations
Tri Lai Bijection Between Catalan Objects
![Page 208: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/208.jpg)
Heaps – Permutations
5
1
1
2
2
3
3
4
4
5
Tri Lai Bijection Between Catalan Objects
![Page 209: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/209.jpg)
Heaps – Permutations
5
1
1
2
2
3
3
4
4
5
Tri Lai Bijection Between Catalan Objects
![Page 210: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/210.jpg)
Heaps – Permutations
A permutation σ is 321-avoiding if there no i < j < k such thatσ(i) > σ(j) > σ(k).
Theorem
Strict heaps are equinumerous to 321-avoiding permutations.
Presentation Topic 1: Permutations with forbidden patterns and Catalannumbers.
Tri Lai Bijection Between Catalan Objects
![Page 211: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/211.jpg)
q-analogs of Catalan Numbers
q-integer [n]q := 1 + q + q2 + . . .+ qn−1
q-factorial [n]q! := [1]q · [2]q · [3]q . . . [n]q[nk
]q
=[n]q!
[n−k]q![k]q!
Tri Lai Bijection Between Catalan Objects
![Page 212: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/212.jpg)
q-analogs of Catalan Numbers
q-integer [n]q := 1 + q + q2 + . . .+ qn−1
q-factorial [n]q! := [1]q · [2]q · [3]q . . . [n]q[nk
]q
=[n]q!
[n−k]q![k]q!
Tri Lai Bijection Between Catalan Objects
![Page 213: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/213.jpg)
q-binomial Coefficients
[a + ba
]q
=?
Tri Lai Bijection Between Catalan Objects
![Page 214: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/214.jpg)
q-analogs of Catalan Numbers
a
b
[a + ba
]q
=?
Tri Lai Bijection Between Catalan Objects
![Page 215: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/215.jpg)
q-analogs of Catalan Numbers
a
b
[a + ba
]q
=?
Tri Lai Bijection Between Catalan Objects
![Page 216: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/216.jpg)
q-analogs of Catalan Numbers
a
b
∑F⊂[a×b]
qarea(F) =
[a + ba
]q
Tri Lai Bijection Between Catalan Objects
![Page 217: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/217.jpg)
q-analogs of Catalan Numbers
a
b
∑F⊂[a×b]
qarea(F) =
[a + ba
]q
Exercise: Prove that the number of k-dimensional subspaces of Fnq is[
nk
]q
.
Tri Lai Bijection Between Catalan Objects
![Page 218: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/218.jpg)
q-analogs of Catalan Numbers
a
b
∑F⊂[a×b]
qarea(F) =
[a + ba
]q
Exercise: Denote by p(j , k , n) the number of integer partitions of n intoat most k parts and each part is at most j . Then∑
n
p(j , k , n)qn =
[j + kk
]q
.
Tri Lai Bijection Between Catalan Objects
![Page 219: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/219.jpg)
q-analogs of Catalan Numbers
A q-analog of Catalan numbers
Cn(q) :=1
[n + 1]q
[2n
n
]q
Tri Lai Bijection Between Catalan Objects
![Page 220: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/220.jpg)
q-analogs of Catalan Numbers
A q-analog of Catalan numbers
Cn(q) :=1
[n + 1]q
[2n
n
]q
Tri Lai Bijection Between Catalan Objects
![Page 221: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/221.jpg)
q-analogs of Catalan Numbers
Cn(q) :=1
[n + 1]q
[2nn
]q
Tri Lai Bijection Between Catalan Objects
![Page 222: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/222.jpg)
q-analogs of Catalan Numbers
3 8 12 18
Cn(q) :=1
[n + 1]q
[2nn
]q
Tri Lai Bijection Between Catalan Objects
![Page 223: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/223.jpg)
q-analogs of Catalan Numbers
maj(w) = 3 + 8 + 12 + 18 = 31
3 8 12 18
Cn(q) :=1
[n + 1]q
[2nn
]q
Tri Lai Bijection Between Catalan Objects
![Page 224: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/224.jpg)
q-analogs of Catalan Numbers
3 8 12 18∑Dyck path w ; |w |=2n
qmaj(w) =1
[n + 1]q
[2nn
]q
Tri Lai Bijection Between Catalan Objects
![Page 225: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/225.jpg)
q-analogs of Catalan Numbers
area(w) = 18
Tri Lai Bijection Between Catalan Objects
![Page 226: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/226.jpg)
q-analogs of Catalan Numbers
∑Dyck path w
qarea(w)t |w |/2
Tri Lai Bijection Between Catalan Objects
![Page 227: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/227.jpg)
q-analogs of Catalan Numbers
Theorem
∑Dyck path w
qarea(w)t |w |/2 =1
1−t
1−tq
1−tq2
1−tq3
1−tq4
1−tq5
1−tq6
1− · · ·
Tri Lai Bijection Between Catalan Objects
![Page 228: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/228.jpg)
Polya q-Catalan Numbers
Tri Lai Bijection Between Catalan Objects
![Page 229: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/229.jpg)
Polya q-Catalan Numbers
Tri Lai Bijection Between Catalan Objects
![Page 230: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/230.jpg)
Polya q-Catalan Numbers
q-Bessel functions
Tri Lai Bijection Between Catalan Objects
![Page 231: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/231.jpg)
(q, t)-Catalan Numbers
Bounce
Tri Lai Bijection Between Catalan Objects
![Page 232: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/232.jpg)
(q, t)-Catalan Numbers
Bounce
Tri Lai Bijection Between Catalan Objects
![Page 233: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/233.jpg)
(q, t)-Catalan Numbers
Bounce
Tri Lai Bijection Between Catalan Objects
![Page 234: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/234.jpg)
(q, t)-Catalan Numbers
Bounce
Tri Lai Bijection Between Catalan Objects
![Page 235: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/235.jpg)
(q, t)-Catalan Numbers
Bounce
Tri Lai Bijection Between Catalan Objects
![Page 236: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/236.jpg)
(q, t)-Catalan Numbers
bounce(w) = 4 + 6 + 9 + 11
012345678910111213
Tri Lai Bijection Between Catalan Objects
![Page 237: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/237.jpg)
(q, t)-Catalan Numbers
bounce(w) = 4 + 6 + 9 + 11 = 30
012345678910111213
Tri Lai Bijection Between Catalan Objects
![Page 238: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/238.jpg)
(q, t)-Catalan Numbers
area and bounce statistics have the same distribution!
Tri Lai Bijection Between Catalan Objects
![Page 239: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/239.jpg)
(q, t)-Catalan Numbers
area and bounce statistics have the same distribution!∑Dyck path w
qarea(w) =∑
Dyck path w
qbounce(w)
Tri Lai Bijection Between Catalan Objects
![Page 240: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/240.jpg)
(q, t)-Catalan Numbers
(q, t)-Catalan Numbers:
Cn(q, t) =∑
Dyck path w ; |w |=2n
qarea(w)tbounce(w)
Tri Lai Bijection Between Catalan Objects
![Page 241: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/241.jpg)
(q, t)-Catalan Numbers
Cn(q, t) =∑
Dyck path w ; |w |=2n
qarea(w)tbounce(w)
This polynomial is symmetric in q, t:
Cn(q, t) = Cn(t, q)
Tri Lai Bijection Between Catalan Objects
![Page 242: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/242.jpg)
(q, t)-Catalan Numbers
Cn(q, t) =∑
Dyck path w ; |w |=2n
qarea(w)tbounce(w)
This polynomial is symmetric in q, t:
Cn(q, t) = Cn(t, q)
There is no bijective proof!
Tri Lai Bijection Between Catalan Objects
![Page 243: Tri Lai - math.unl.edutlai3/Catalan Objects.pdf · Catalan Numbers C n = 1 n+1 2n n The number offull binary treewith2n + 1vertices (i.e., n internal vertices). The number oftriangulationsof](https://reader033.vdocuments.net/reader033/viewer/2022041609/5e36a19ca6c1c809f93fe394/html5/thumbnails/243.jpg)
Original definition of (q, t)-Catalan Numbers
aa’
l’
l
A. Garsia, M. Haiman (1994)
Cn(q, t) =∑λ`n
t2∑
c∈λ lq2∑
c∈λ a(1− t)(1− q)∏
c∈λ(1− qa′t l
′)∑
c∈λ qa′t l
′∏c∈λ(qa − t l+1)(t l − qa+1)
Presentation Topic 2: (q, t)-Catalan numbers.Presentation Topic 3: ‘Kepler Towers’ and Catalan numbers.
Tri Lai Bijection Between Catalan Objects