a funny thing happened on the way to steppenwolf...
TRANSCRIPT
![Page 1: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/1.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A funny thing happened on the way toSteppenwolf Theatre. . .
from lattice paths to polytopes and Hopf algebras
T. Kyle Petersen
DePaul UniversityDepartment of Mathematical Sciences
MathFestMadison, WIAugust 2012
![Page 2: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/2.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Combinatorics, geometry, and an algebra of paths
1 Combinatorics: Walking to Steppenwolf
2 Geometry: Tamari poset/associahedron
3 Algebra: Loday-Ronco
![Page 3: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/3.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
The problem
![Page 4: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/4.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
The problem
![Page 5: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/5.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
Let Cn denote the number of paths of length 2n
![Page 6: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/6.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
Let Cn denote the number of paths of length 2n
C0 = 1 :
![Page 7: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/7.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
Let Cn denote the number of paths of length 2n
C0 = 1 :
C1 = 1 :
![Page 8: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/8.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
Let Cn denote the number of paths of length 2n
C0 = 1 :
C1 = 1 :
C2 = 2 :
![Page 9: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/9.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
Let Cn denote the number of paths of length 2n
C0 = 1 :
C1 = 1 :
C2 = 2 :
C3 = 5 :
![Page 10: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/10.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C4 = 14 :
![Page 11: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/11.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5?
![Page 12: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/12.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5?
Point of second-to-last return
![Page 13: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/13.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5?
Point of second-to-last return
![Page 14: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/14.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
![Page 15: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/15.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
![Page 16: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/16.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
![Page 17: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/17.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
![Page 18: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/18.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
![Page 19: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/19.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
![Page 20: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/20.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
![Page 21: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/21.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
+
![Page 22: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/22.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
+
C3 · C1
![Page 23: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/23.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
+
C3 · C1
+
![Page 24: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/24.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
+
C3 · C1
+
C4 · C0
![Page 25: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/25.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Counting “Clybourn paths”
C5 =
C0 · C4
+
C1 · C3
+
C2 · C2
+
C3 · C1
+
C4 · C0
= 1 · 14 + 1 · 5 + 2 · 2 + 5 · 1 + 14 · 1 = 42
![Page 26: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/26.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
![Page 27: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/27.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . .
![Page 28: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/28.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1
n + 1
(
2n
n
)
, . . .
![Page 29: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/29.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1
n + 1
(
2n
n
)
, . . .
Also counts:
balanced parenthesizations (()(()))
![Page 30: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/30.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1
n + 1
(
2n
n
)
, . . .
Also counts:
balanced parenthesizations (()(()))
planar binary trees
![Page 31: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/31.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1
n + 1
(
2n
n
)
, . . .
Also counts:
balanced parenthesizations (()(()))
planar binary trees
noncrossing partitions:• •
••
![Page 32: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/32.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Catalan numbers
C0 = 1, Cn+1 =n∑
i=0
CiCn−i
1, 1, 2, 5, 14, 42, 132, 429, 1430, . . . ,Cn =1
n + 1
(
2n
n
)
, . . .
Also counts:
balanced parenthesizations (()(()))
planar binary trees
noncrossing partitions:• •
••
and about 200 other sets of things. . .
![Page 33: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/33.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Combinatorics, geometry, and an algebra of paths
1 Combinatorics: Walking to Steppenwolf
2 Geometry: Tamari poset/associahedron
3 Algebra: Loday-Ronco
![Page 34: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/34.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
![Page 35: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/35.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
![Page 36: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/36.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
→
![Page 37: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/37.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
![Page 38: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/38.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
![Page 39: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/39.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
→
![Page 40: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/40.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
![Page 41: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/41.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
![Page 42: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/42.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
→
![Page 43: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/43.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
![Page 44: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/44.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
![Page 45: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/45.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
→
![Page 46: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/46.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
n = 0
![Page 47: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/47.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
n = 0 n = 1
![Page 48: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/48.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
n = 0 n = 1 n = 2
![Page 49: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/49.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A local transformation
n = 0 n = 1 n = 2 n = 3
![Page 50: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/50.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Tamari poset (Associahedron)
n = 4
![Page 51: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/51.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Combinatorics, geometry, and an algebra of paths
1 Combinatorics: Walking to Steppenwolf
2 Geometry: Tamari poset/associahedron
3 Algebra: Loday-Ronco
![Page 52: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/52.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Jean-Louis Loday: “the integers as molecules”
0 1 2 3 4
![Page 53: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/53.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A meditation
We can do arithmetic at the molecular level. Can we do it at theatomic level?
![Page 54: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/54.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
The wedge operation
p ∨ q:
∨ =
![Page 55: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/55.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
The wedge operation
p ∨ q:
∨ =
![Page 56: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/56.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
The wedge operation
p ∨ q:
∨ =
(Recall the role the wedge played in the Catalan identity)
![Page 57: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/57.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Decomposing paths
p = pℓ ∨ pr (unique decomposition)
= ∨
![Page 58: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/58.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Decomposing paths
p = pℓ ∨ pr (unique decomposition)
= ∨
= ∨ = ( ∨ ) ∨
![Page 59: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/59.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Decomposing paths
p = pℓ ∨ pr (unique decomposition)
= ∨
= ∨ = ( ∨ ) ∨
= ∨ = ∨ ( ∨ )
![Page 60: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/60.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Defining addition
Goal: define addition for paths,
p + q,
in a way that generalizes addition for integers
![Page 61: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/61.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Defining addition
Goal: define addition for paths,
p + q,
in a way that generalizes addition for integers
p + 0 = 0+ p = p
![Page 62: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/62.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Defining addition
Goal: define addition for paths,
p + q,
in a way that generalizes addition for integers
p + 0 = 0+ p = p
for p, q 6= 0, p + q = (p ⊣ q) ∪ (p ⊢ q)
![Page 63: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/63.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Defining addition
Goal: define addition for paths,
p + q,
in a way that generalizes addition for integers
p + 0 = 0+ p = p
for p, q 6= 0, p + q = (p ⊣ q) ∪ (p ⊢ q)
now recursively,
p ⊣ q = pℓ ∨ (pr + q) and p ⊢ q = (p + qℓ) ∨ qr
![Page 64: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/64.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
![Page 65: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/65.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
1
![Page 66: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/66.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
1 1
+
![Page 67: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/67.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
1 1
+
2
=
![Page 68: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/68.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
Split the plus sign: + = ⊣ ∪ ⊢
+ =(
⊣)
⋃
(
⊢)
![Page 69: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/69.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
Split the plus sign: + = ⊣ ∪ ⊢
+ =(
⊣)
⋃
(
⊢)
=(
∨(
+))
⋃
((
+)
∨)
![Page 70: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/70.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
Split the plus sign: + = ⊣ ∪ ⊢
+ =(
⊣)
⋃
(
⊢)
=(
∨(
+))
⋃
((
+)
∨)
=(
∨)
⋃
(
∨)
![Page 71: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/71.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 1 = 2
Split the plus sign: + = ⊣ ∪ ⊢
+ =(
⊣)
⋃
(
⊢)
=(
∨(
+))
⋃
((
+)
∨)
=(
∨)
⋃
(
∨)
=⋃
![Page 72: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/72.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
![Page 73: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/73.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
Want:
1
+
2
=
3
![Page 74: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/74.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
+ =
(
+
)
⋃
(
+
)
![Page 75: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/75.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
+ =
(
+
)
⋃
(
+
)
=
(
⊣
)
⋃
(
⊢
)
⋃
(
⊣
)
⋃
(
⊢
)
![Page 76: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/76.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
![Page 77: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/77.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨ =
![Page 78: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/78.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨ =
⊣ = ∨
(
+
)
=
![Page 79: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/79.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨ =
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨
![Page 80: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/80.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨ =
⊣ = ∨
(
+
)
=
⊢ = ∨
![Page 81: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/81.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Why 1+ 2 = 3
⊣ = ∨
(
+
)
=
⊢ =(
+)
∨ =
⊣ = ∨
(
+
)
=
⊢ = ∨ =⋃
![Page 82: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/82.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Surprise!
Theorem
If a+ b = c (as nonnegative integers), then
a+ b = c
(as Tamari lattices)
![Page 83: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/83.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
A better way to add
Can we add two paths more simply?
![Page 84: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/84.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
![Page 85: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/85.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
![Page 86: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/86.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
![Page 87: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/87.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
\
![Page 88: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/88.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
�
�
![Page 89: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/89.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
�
�
/
![Page 90: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/90.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
+
�
�
�
![Page 91: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/91.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
Theorem (Loday-Ronco)
For any paths p and q,
p + q =⋃
p/q≤r≤p\q
r
![Page 92: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/92.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
Theorem (Loday-Ronco)
For any paths p and q,
p + q =⋃
p/q≤r≤p\q
r
There is also a multiplication that refines ordinary integermultiplication. . .
![Page 93: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/93.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Adding paths and the Tamari poset
Theorem (Loday-Ronco)
For any paths p and q,
p + q =⋃
p/q≤r≤p\q
r
There is also a multiplication that refines ordinary integermultiplication. . . the Loday-Ronco algebra is the “free dendriformalgebra on one generator” and a “combinatorial Hopf algebra”
![Page 94: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/94.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Food for thought
We can now do arithmetic:
![Page 95: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/95.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Food for thought
We can now do arithmetic:
at the molecular level (whole numbers)
![Page 96: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/96.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Food for thought
We can now do arithmetic:
at the molecular level (whole numbers)
at the atomic level (paths)
![Page 97: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/97.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
Food for thought
We can now do arithmetic:
at the molecular level (whole numbers)
at the atomic level (paths)
What about the subatomic level?
![Page 98: A funny thing happened on the way to Steppenwolf Theatremath.depaul.edu/tpeter21/MathFestTalk.pdf · Combinatorics: WalkingtoSteppenwolf Geometry: Tamariposet/associahedron Algebra:](https://reader031.vdocuments.net/reader031/viewer/2022020303/5b5268837f8b9a56588d49b2/html5/thumbnails/98.jpg)
Combinatorics: Walking to Steppenwolf Geometry: Tamari poset/associahedron Algebra: Loday-Ronco
References
M. Aguiar and F. Sottile, Structure of the Loday-Ronco Hopfalgebra of trees. J. Algebra 295 (2006), 473–511.
J.-L. Loday, Dichotomy of the addition of natural numbers. inAssociahedra, Tamari Lattices and Related Structures,Progress in Math. 299, Birkhauser (2012). (alsoarXiv:1108.6238)
J.-L. Loday, Arithmetree. J. Algebra 258 (2002), 275–309.
J.-L. Loday and M. Ronco, Hopf algebra of the planar binarytrees. Adv. Math. 139 (1998), 293–309