Enumerative Combinatorics with Fillings of Polyominoes Catherine Yan Texas A&M University

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� Enumerative Combinatorics: counting the number of elements of a finite set S, e.g., permutations, words, matchings, set partitions, integer partitions, paths, graphs …

� Combinatorial statistics f:  S  !  N          e.g.  cycle,  length,  block,  degree…                    

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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

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# noncrossing matchings of [2n] = # nonnesting matchings of [2n]

k-crossings/nestings

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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

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

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

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].

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Corollary.

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

A = B

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.

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

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].

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1 2 3 4 5 6 7

Filling of the staircase

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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

01-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

1 1

1

1

1

1 1

Ferrers diagram

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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

Generalized triangulation of n-gon

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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

avoid

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!

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try to avoid

Stack polyominoes

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[Jonsson’05, Jonsson & Welker’07]: The number of 01- 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’11]: The number of 01- 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

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 ◦ Major index ◦ Connections to other objects

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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.

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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

Mixed statistics

� Mix by colors of the cell -- Bicolor the rows of M: (S, M-S) � Mix by sign of the cell The symmetric joint distribution is unchanged.

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+

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 February 2014

It applies to other patterns.

� Both papers compared patterns Jk and

� One can get more Wilf-equivalent pairs.

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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

equa-distributed 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.

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

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

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w =w1 …wn-1 a

w1… wn-1 u1…  un-­‐1   v1…  vn-­‐1

v1… vn-­‐1 a

Φ γa

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

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•  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))

And more…

� Lattice path counting and descents in Ferrers diagrams [Song & Y’12]

�  Integer sequences � Pattern avoidance and appearances � Poset, P-partitions …

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Relation to other areas…

�  Free probability－ noncrossing diagrams

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� Graph optimization and layout: stack, queue, and arch

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•  Combinatorial computational biology: RNA pseudo knot structures

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Thank you very much!

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