the symmetry between crossings and nestings in combinatorics

The Symmetry between Crossings and Nestings in Combinatorics

Catherine Yan

Texas A&M University

College Station, TX, USA

Outline

� Three pair of combinatorial statistics

1. Chains of length 2

2. Longest chains

3. Mixed chains

� One Combinatorial Model:

fillings of moon polyominoes

May 19--22 DMEC 2011, Taipei

Part I: The first pair

Permutations: inversion v.s. coinversion

π=624153

� inversion: {(i - j): i > j } coinversion: {(i - j): i < j }

inv(π)= 9 : { 62, 64, 61, 65, 63, 21, 41, 43, 53}

coinv(π)=6: { 24, 25, 23, 45, 15, 13 }

∑π p

inv(π)qcoinv(π) =∏n

i=1(pi + pi−1q + · · ·+ pqi−1 + qi)


On words over { 1n1, 2n2,…, knk }

� A word is an arrangement of 1n1, 2n2, …, knk


∑

W

pinv(W )qcoinv(W ) =

(n

n1, n2, ..., nk

)

p,q

where the p,q-integer is

[k]p,q= pk-1 +pk-2q + …+ pqk-2 +qk-1

crossings and nestingsin matchings

� (cr2, ne2) has a symmetric joint

distribution.

e.g.

∑M pcr2(M) q

ne2(M)

=p2+2pq+q2

=[2]p,q [2]p,q


Partitions on {1,2,…, n}

� A graphical representation

Π= {1,4,5} {2,6} {3}


1 2 3 4 5 6

Four types: opener 1, 2, closer 5, 6singleton 3 transient 4

Set partitions with the same pattern

� (cr2, ne2) has a symmetric

joint distribution.

e.g.

∑M pcr2(M) qne2(M)

=p2+2pq+q2

=[2]p,q [2]p,q


Filling of the staircase

1 2 3 4 5 6 7 8

8 7

6 5

4 3

2

Crossing: anti-identity submatrix

(NE-chain)

Nesting: identity submatrix

(SE-chain)

1


Filling of a Ferrers diagram

Put three 1’s in it, such that

each row has one 1.

We are interested in the numbers of

1

1 1

1and


� The bivariate G.F.

4 + p + q + 2pq + p2 + q2

� However, NOT symmetric for


When do we still have the symmetry?

A General Model: moon polyominoes

� Polyomino: a finite set of square cells

� Moon polyomino: ◦ Convex

◦ intersection-free (no skew shape)


Fillings of moon polyominoes

� Assign 0 or 1 to each square

inversion

coinversion

1

1

1

1

NE-chains

SE-chains

1

1

1

1

1

1

11

1

1


Kasraoui’s Theorem

� The pair (ne2, se2) has a symmetric distribution over the set F(M, t), where

M = moon polyomino,

F(M, t) = {01-fillings with arbitrary row sum t, and every column has at most one 1 }

In fact,


∑

M∈F (M,t)

pne2(M)qse2(M) =∏

i

(hi

ti

)

p,q

Part II: the second pair

� Permutations:

732816549 (increasing subsequence)

732816549 (decreasing subsequence)

is(w) = | longest i. s.| = 3

ds(w)= | longest d. s.| = 4


Higher crossings/nestings on matchings

#Matchings on [2n] with no 3-crossings

= # Matchings on [2n] with no 3-nestings

= # Pairs of non-crossing Dyck paths

#Matchings on [2n] with no 3-crossings

= # Matchings on [2n] with no 3-nestings

= # Pairs of non-crossing Dyck paths


k-crossings/nestings

#Matchings on [2n] with no k-crossings

= # Matchings on [2n] with no k-nestings

#Matchings on [2n] with no k-crossings

= # Matchings on [2n] with no k-nestings


Crossing and nesting number

� For a matching M, cr(M)=max{ k: M has a k-crossing } ne(M)=max{k: M has a k-nesting }

Goal: symmetry between cr and ne

# k-crossing and # k-nesting: may not be symmetric

How about “maximal crossing” and “maximal nesting”?


Main result on Matchings

� Theorem [Chen, Deng, Du, Stanley, Y]

Let fn(i,j) = # matchings M on [2n] with cr(M)=i and ne(M)=j. Then

fn(i,j) = fn(j, i)

� Corollary. # matchings M without k-crossing equals # matchings without k-nesting.


Idea:

� Oscillating tableau: a sequence of Ferrersdiagrams ∅=λ0, λ1, …, λ2n =∅ s.t.

λλλλi= λλλλi-1 +/ -

∅

∅


� Theorem. [Stanley & Sundaram]

There is a bijection between matchings and oscillating tableaux.

� Cor. # Oscillating tableaux of length 2n

is (2n-1)!!


Example


1 2 43 5 6 7 8

Theorem [CDDSY]: φ: M[2n] OT(2n)

Then

cr(M) = Most #rows in any partition of φ(M)

ne(M)=Most #columns in any partition of φ(M)


Theorem [CDDSY]: φ: M[2n] OT(2n)

Then

cr(M) = Most #rows in any partition of φ(M)

ne(M)=Most #columns in any partition of φ(M)


Symmetry of cr(M) and ne(M)

λ λ’

cr(M) ne(M’)

Set Partitions

� A graphical representation

π= {1,4, 5, 7} {2,6} {3}


1 2 3 4 5 6 7

Vacillating tableau


1 2 3 4 5 6 7

∅

VT: a sequence of Ferrers diagrams∅=λ0, λ1, …, λ2n =∅ s.t.

λλλλ2i+1= λλλλ2i or λλλλ2i + ��λλλλ2i =λλλλ2i-1 or λλλλ2i-1-��

Filling of the staircase

1 2 3 4 5 6 7 8

8 7

6 5

4 3

2

Crossing: anti-identity submatrix

(NE-chain)

Nesting: identity submatrix

(SE-chain)

1


An extension to Ferrers Diagram

0-1 filling of any Ferrers diagram F

Every row/column has at most one 1.

NE-chain Jk SE-chain Ik


1

11

1

1

1

1

1

Ferrer’s shape

Theorem.[Krattenthaler]

Given F, n, s, t. Then

#N(F; n; NE=s, SE=t)

= #N(F; n; NE=t, SE=s}


Multi-graph


1

2

34

5

1 2 3 4

5 4

3 2

Theorem. [de Mier] Fix the (left-right) degree

sequences, order and size.

#{ G: k-noncrossing } = #{G: k-nonnesting }

K-crossing:K-nesting:

2

1 1

1 1

1 1

Multi-graph is equivalent to filling of Ferrers diagrams


l1 r1l3 r2 l4 r3 r4l2

l1 l2 l3 l4

r4

r3

r2

r1 1

1

1

1

Generalized triangulation of n-gon


1 2

3

4

56

7

8

1

23

4

5

6

7

7 6 5 4 3 28

k-triangulation:

no k+1 diagonals that are mutually intersecting

avoid

Results about k-triangulation

� [Capoyleas & Pach, 1992]

k-triangulations of an n-gon has at most

k(2n-2k-1) lines.

� [Dress, Koolen & Moulton, 2002]

maximal k-triangulation always has

k(2n-2k-1) lines

� [Jonsson 2005] #maximal k-triangulations

=a determinant of Catalan numbers.


Catalan number implies symmetry!


1

23

4

5

6

7

7 6 5 4 3 28Instead of

try to avoid

Stack polyominoes


[Jonsson & Welker]: The number of 0-1 fillings

with m nonzero entries that avoid the matrix Jk depends

only on the size of the columns, not on the ordering of

the columns.

Moon polyominoes


[Rubey, 2007]: The number of 0-1 fillings



the columns.

Moon polyominoes


[Rubey, 2007]: The number of 0-1 fillings



the columns.

Symmetry between Ik and Jk:

flipping the moon polyomino

Part III: the mixed pair

Permutation again: π=624153

� inversion: {(i-j): i >j } coinversion: {(i-j): i<j}

inv(π)= 9 coinv(π)=6

� A permutation statistic is a Mahonian if it has the same distribution as inv, e.g. the major index…


A variation of inversion

[Chebikin’08] π=624153

α(π) = #| {(i, j): i>j}| + #|{(i, j): i<j}|

e.g. 62, 64, 61, 65, 63, 41, 43, 53, 24, 25, 23, 15, 13

α(π)=13

β(π)=n(n-1)/2-α(π) = 2

Actually, the bicoloring can be ARBITRARY!

(α(π), β(π)) has the same distribution as (inv(π), coinv(π))


Crossing and nesting on matching

Consider a combination of cr2 and ne2

std1:

std2:

and

and


Crossings and nestings

std1, std2

1 1

2 0

1 1

0 2

Observation: (std1, std2) has the same

symmetric distribution as

(cr2, ne2) over all matchings

with fixed left and right endpoints.

Similar observation for set partitions.


How about moon polyominoes


Recall Kasraoui’s Theorem

� The pair (ne2, se2) has a symmetric distribution of the set F(M, t), where

where

F(M, t) = {01-fillings with arbitrary row sum t, and every column has at most one 1 }


Four mixed statistics

� Bicolor the rows of M: (S, M-S)

1

11

1top-mixed statistic α(S,M): and

bottom-mixed statistic β(S,M):

1

1

1

1and


Four mixed statistics

� Bicolor the columns of M: (T, M-T)

1

11

1left-mixed statistic

γ(T,M): and

right-mixed statistic

δ(T,M): 1

1

1

1and


Symmetry on mixed statistics

Theorem. [Chen, Wang, Y, Zhao]

Let λ(A,M) be any of these four mixed statistics. Then the joint distribution of the pair

(λ(A, M), λ (M-A, M))

is symmetric and independent of the subset A.

Note: (λ(∅, M), λ(M, M)) = (se2(M), ne2(M))

(λ(M, M), λ(∅, M)) = (ne2(M), se2(M))


Why?

� Rows of M can be ordered from small to large

� Filling of each row can be written as a composition

� Local transformations in rectangles


Summary and Remarks



2. Longest chains

3. Mixed chains


Summary and Remarks



2. Longest chains

3. Mixed chains

� One Combinatorial Model:

fillings of moon polyominoes


Projects on-going

� Distribution of (cr2, ne2) generated from a root;

� Major index and other Mahonian statistics

� Crossings and nestings on matchings over binary string

� Joint distribution of Euler-Mahonian statistics

� Relations with geometry, representation, Coexter groups, …


Thank you very much!

Collaborators: W. Chen, A. de Mier, E. Deng, R. Du,

I. Gessel, S. Poznanovik, R. Stanley, A.Wang, S. Wu,

A. Yang, A. Zhao

References available at:

http://www.math.tamu.edu/~cyan


the symmetry between crossings and nestings in combinatorics

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