The Symmetry between Crossings and Nestings in Combinatorics
Transcript of 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)
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On words over { 1n1, 2n2,…, knk }
� A word is an arrangement of 1n1, 2n2, …, knk
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∑
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
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Partitions on {1,2,…, n}
� A graphical representation
Π= {1,4,5} {2,6} {3}
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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
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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
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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
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� The bivariate G.F.
4 + p + q + 2pq + p2 + q2
� However, NOT symmetric for
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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)
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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
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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,
May 19--22 DMEC 2011, Taipei
∑
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
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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
May 19--22 DMEC 2011, Taipei
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
May 19--22 DMEC 2011, Taipei
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”?
May 19--22 DMEC 2011, Taipei
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.
May 19--22 DMEC 2011, Taipei
Idea:
� Oscillating tableau: a sequence of Ferrersdiagrams ∅=λ0, λ1, …, λ2n =∅ s.t.
λλλλi= λλλλi-1 +/ -
∅
∅
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� Theorem. [Stanley & Sundaram]
There is a bijection between matchings and oscillating tableaux.
� Cor. # Oscillating tableaux of length 2n
is (2n-1)!!
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� Theorem. [Stanley & Sundaram]
There is a bijection between matchings and oscillating tableaux.
� Cor. # Oscillating tableaux of length 2n
is (2n-1)!!
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Example
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1 2 43 5 6 7 8
Example
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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)
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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)
May 19--22 DMEC 2011, Taipei
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}
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1 2 3 4 5 6 7
Vacillating tableau
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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
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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
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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}
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Multi-graph
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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
May 19--22 DMEC 2011, Taipei
l1 r1l3 r2 l4 r3 r4l2
l1 l2 l3 l4
r4
r3
r2
r1 1
1
1
1
Generalized triangulation of n-gon
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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.
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Catalan number implies symmetry!
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1
23
4
5
6
7
7 6 5 4 3 28Instead of
try to avoid
Stack polyominoes
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[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
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[Rubey, 2007]: 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
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[Rubey, 2007]: 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.
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…
May 19--22 DMEC 2011, Taipei
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))
May 19--22 DMEC 2011, Taipei
Crossing and nesting on matching
Consider a combination of cr2 and ne2
std1:
std2:
and
and
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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.
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How about moon polyominoes
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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 }
May 19--22 DMEC 2011, Taipei
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
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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
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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))
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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
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Summary and Remarks
� Three pair of combinatorial statistics
1. Chains of length 2
2. Longest chains
3. Mixed chains
May 19--22 DMEC 2011, Taipei
Summary and Remarks
� 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
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, …
May 19--22 DMEC 2011, Taipei
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
May 19--22 DMEC 2011, Taipei