Enumerative Combinatorics with Fillings of Polyominoes

Catherine Yan Texas A&M Univesrity GSU, October, 2014

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Outline 1.  Symmetry of the longest chains ◦  Subsequences in permutations and words ◦  Crossings and nestings in matchings and graphs ◦  A new model: fillings of moon polyominoes

2.  Combinatorics of Fillings of Moon polyominoes ◦  Northeast and southeast chains ◦  Forbidden patterns ◦  Transformations ◦  Connections to other objects

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Part I: Symmetry of the longest chains � Permutations: 732816549 (increasing subsequence) 732816549 (decreasing subsequence) is(w) = | longest i. s.| = 3 ds(w)= | longest d. s.| = 4

�  [Deift, Baik & Johansson’99] Asymptotic distribution of is(w) and ds(w).

�  is(w) and ds(w) are symmetric.

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Crossings and nestings in matchings of [2n]

�  (cr2, ne2) are symmetric! e.g. cr2 ne2

2 0 1 1 1 1 0 2

# noncrossing matchings of [2n] = # nonnesting matchings of [2n] = nth Catalan number

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k-crossings/nestings

Conjecture: #Matchings of [2n] with no k-crossings = # Matchings of [2n] with no k-nestings

Theorem [Chen, Deng, Du] # matchings of [2n] with no 3-crossings = # matching of [2n] with no 3-nestings = # pairs of noncrossing Dyck paths

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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: not symmetric How about “maximal crossing” and “maximal nesting”?

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Main result on Matchings Theorem [Chen, Deng, Du, Stanley & Y, 07] The pair (cr(M), ne(M)) has a symmetric joint distribution over all matchings on [2n].

Corollary.

# matchings with no k-crossing = # matchings with no k-nesting

A = B

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Idea:

� Oscillating tableau: a sequence of Ferrers diagrams ;=λ0, λ1, …, λ2n =; s.t.

λi= λi-1 +/ -

;

;

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Theorem [Stanley & Sundaram’90] There is a bijection between matchings of [2n] and oscillating tableaux of length 2n.

�  It is realized by using standard Young tableaux and applying the RSK algorithm.

Theorem [CDDSY] Taking conjugation in the tableaux exchanges cr(M) and ne(M).

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Set Partitions of [n]

� A graphical representation π= {1,4, 5, 7} {2,6} {3} � Theorem. [CDDSY] (cr(¼), ne(¼)) has a symmetric distribution over all partitions of [n].

1 2 3 4 5 6 7

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Filling of the triangular board

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

01-filling of any Ferrers diagram F Every row/column has at most one 1. NE-chain Jk SE-chain Ik

1

1 1

1

1

1

1 1

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Ferrers diagram

NE(F) = |longest NE chain| SE (F) = |longest SE chain|

[Krattenthaler’06] Given a Ferrers diagram F and an integer n, then (NE(F), SE(F)) has a symmetric distribution over 01-fillings of F with n 1’s..

1

1

1

1

1

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Generalized triangulation of n-gon

1 2

3

4

5 6

7

8

1

2 3 4 5 6 7

7 6 5 4 3 2 8

k-triangulation: no k+1 diagonals that are mutually intersecting

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Results about k-triangulation �  [Capoyleas & Pach’92] k-triangulations of an n-gon has at most k(2n-2k-1) lines. �  [Dress, Koolen & Moulton’02] maximal k-triangulation always has k(2n-2k-1) lines �  [Jonsson’05] #maximal k-triangulations =a determinant of Catalan numbers.

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Catalan number implies symmetry!

try to avoid

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Stack polyominoes

[Jonsson’05, Jonsson & Welker’07]: F01(L, n, ne < k ) = F01 (L, n, se < k) where n is the number of ones in the filling.

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Moon polyominoes

[Rubey’11]: F(M, n, ne<k ) = F(M, n, se <k ) And F01(M, n, ne< k) = F01 (M, n, se <k)

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The General Model: fillings of 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 an integer to each square

1 1

1

1

1

1

11

1

1

Permuta-tions Words Matchings Set partitions

Graphs Ferrers diagram Stack polyomino Moon polyomino

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Part II: Combinatorics of fillings of moon polyominoes

◦ Northeast and southeast chains ◦ Forbidden patterns ◦ Transformations ◦ Connections to other objects

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The model is general: Example 1. Chains of length 2 Permutation: inversion and coinversion

¼=624153 �  inversion: {(i - j): i > j } coinversion: {(i - j): i < j } inv(¼)= 9 : { 62, 64, 61, 65, 63, 21, 41, 43, 53}

coinv(¼)=6: { 24, 25, 23, 45, 15, 13 }

P¼ p

i n v (¼) qcoi n v (¼) =A =P

i BiA+B = cA+B = CA = B

A = B

where [k]p.q is the (p,q)-integer pk-1+pk-2q + … + pqk-2 + qk-1.

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On words over { 1n1, 2n2,…, knk }

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

� Similar results for

◦  Matchings [de Sainte-Catherine’83] ◦  Set partitions [Kasraoui & Zeng’06] ◦  Linked partitions [Chen, Wu & Y’ 08] ◦  Crossing and alignment for permutations [Corteel’07]

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Theorem [Kasraoui’10] The pair (ne2, se2) has a symmetric joint distribution over the set of 01-fillings of a moon polyomino with any given column sum.

X

M 2 F (M ;t)

pn e2(M ) qs e2(M ) =Y

i

µhiti

¶

p;q

X

M 2 F (M ;t)

pn e2(M ) qs e2(M ) =Y

i

µhiti

¶

p;q

a+ b = ca+ b = c

inv(¼) coinv(¼)

1 1

1 1

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various mixed statistics

� Bicolor the rows of M and mixed by the position of the top cell/ bottom cell

1 1 1

1 top-mixed statistic ®(S,M): and

bottom-mixed statistic ¯(S,M):

1 1

1 1 and

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� Mix by the charge of a corner cell

1 1 1

1 Positive chains and

Negative chains 1 1

1 1 and

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Symmetry on mixed statistics

Theorem. [Chen, Wang, Y, Zhao’10; Wang &Y’13 ] Let ¸(A) be the number of any of the mixed statistics. (Hence ¸(M-A) is the number of remaining 2-chains. ) Then the joint

distribution of the pair (¸(A), ¸ (M-A)) is always symmetric and independent of the subset A.

Note: (¸(;), ¸(M)) = (se2(M), ne2(M)) (¸(M), ¸(;)) = (ne2(M), se2(M))

Special case for permutations: Chebikin’08.

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The model is special enough!

Many things happen inside rectangles!

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Example 2: k-noncrossing vs k-nonnesting Problem: # fillings with no k-crossing = # fillings with no k-nesting Method: Start with a filling with no k-crossing, then replace every appearance of k-nesting by other patterns. �  [Backelin, West, Xin’07] for 01-fillings of

Ferrers diagrams �  [de Mier’07] for multi-graphs with fixed

degree sequences 30

It applies to other patterns.

� Both papers compared patterns Jk and

� One can get more Wilf-equivalent pairs.

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Applies to symmetric fillings �  [Bousquet-Melou, Steingrimsson’05]

symmetric 01-fillings of symmetric Ferrers diagrams – involution

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Example 3. The major index

�  For a word a1 a2 … an , a descent is a position i such that ai > ai+1.

�  maj(w) = ∑ { i : i 2 DES(w) }. �  [MacMahon’1916] The major index is

equadistributed to inv(w) over words.

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Example 3. The major index

�  For a word a1 a2 … an , a descent is a position i such that ai > ai+1.

�  maj(w) = ∑ { i : i 2 DES(w) }. �  [MacMahon’1916] The major index is

equadistributed to inv(w) over words.

[Chen, Poznanovik, Y & Yang’ 10] The major index can be extended to 01-fillings of moon polyominoes, which has the same distribution as ne2.

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Foata’s map Φ with inv(Φ(w))=maj(w)

� Recursive Definition: ◦  If w has length 1, Φ(w)=w.

◦  Otherwise, w= w’ a, then Φ(w) = γa(Φ (w’)) a

w =w1 …wn-1 a

w1… wn-1 u1…  un-­‐1   v1…  vn-­‐1

v1… vn-­‐1 a

Φ γa

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Many transformations!

�  [CPYY] Foata-type transformations can be defined on fillings of left-aligned stack polyominoes which carry maj to ne2

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From polyomino to polyomino

•  Bijection f from fillings of M to fillings of N s.t. maj(F) =maj(f(F)) •  Bejection g from fillings of M to fillings of N s.t. ne2(F) = ne2(g(F))

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And more… � Lattice path counting and descents in

Ferrers diagrams � Rook placement with restrictions � Pattern avoidance and appearances � Poset, P-partitions �  Simplicial complexes/Schubert

polynomials …

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Relation to other areas…

�  Free probability－ noncrossing diagrams

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� Crossings appear in the combinatorial interpretations of ◦ Mixed moments of random variables ◦ Moments of orthogonal polynomials ◦  Linearization coefficients …

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� Graph optimization and layout: A partition of the edges into k-sets of non-crossing (non-nesting) edges

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Stacks and Queues

�  Stack-number: minimum k such that there is a total order of the vertices with which G has a k-stack layout

� Queue-number: minimum k such that there is a total order of the vertices with which G has a k-queue layout

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•  Combinatorial computational biology: RNA pseudo knot structures

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T H A N K

Y O U

V E R Y

M U C H !