Enumerative Combinatorics with Fillings of Polyominoes Catherine Yan Texas A&M Univesrity GSU, October, 204
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Outline. 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 3
Part I: Symmetry of the longest chains Permutations: 73286549 (increasing subsequence) 73286549 (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. 4
Crossings and nestings in matchings of [2n] (cr 2, ne 2 ) are symmetric! e.g. cr 2 ne 2 2 0 0 2 # noncrossing matchings of [2n] = # nonnesting matchings of [2n] = nth Catalan number 5
k-crossings/nestings Theorem [Chen, Deng, Du] # matchings of [2n] with no 3-crossings = # matching of [2n] with no 3-nestings = # pairs of noncrossing Dyck paths Conjecture: #Matchings of [2n] with no k-crossings = # Matchings of [2n] with no k-nestings 6
Crossing and nesting number # k-crossing and # k-nesting: not symmetric How about maximal crossing and maximal nesting? 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 7
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 8
Idea: Oscillating tableau: a sequence of Ferrers diagrams ;=λ 0, λ,, λ 2n =; s.t. λ i = λ i- +/ - ; ; 9
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). 0
Set Partitions of [n] A graphical representation π= {,4, 5, 7} {2,6} {3} 2 3 4 5 6 7 Theorem. [CDDSY] (cr(¼), ne(¼)) has a symmetric distribution over all partitions of [n].
Filling of the triangular board 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) 2
An extension to Ferrers diagram 0-filling of any Ferrers diagram F Every row/column has at most one. NE-chain J k SE-chain I k 3
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 0-fillings of F with n s.. 4
Generalized triangulation of n-gon 2 8 7 6 5 4 3 2 8 7 3 4 2 3 4 5 6 7 6 5 k-triangulation: no k+ diagonals that are mutually intersecting 5
Results about k-triangulation [Capoyleas & Pach 92] k-triangulations of an n-gon has at most k(2n-2k-) lines. [Dress, Koolen & Moulton 02] maximal k-triangulation always has k(2n-2k-) lines [Jonsson 05] #maximal k-triangulations =a determinant of Catalan numbers. 6
Catalan number implies symmetry! try to avoid 7
Stack polyominoes [Jonsson 05, Jonsson & Welker 07]: F 0 (L, n, ne < k ) = F 0 (L, n, se < k) where n is the number of ones in the filling. 8
Moon polyominoes [Rubey ]: F(M, n, ne<k ) = F(M, n, se <k ) And F 0 (M, n, ne< k) = F 0 (M, n, se <k) 9
The General Model: fillings of moon polyominoes Polyomino: a finite set of square cells Moon polyomino: Convex intersection-free (no skew shape) 20
Fillings of moon polyominoes Assign an integer to each square Permuta -tions Words Matchings Set partitions Graphs Ferrers diagram Stack polyomino Moon polyomino 2
Part II: Combinatorics of fillings of moon polyominoes Northeast and southeast chains Forbidden patterns Transformations Connections to other objects 22
The model is general: Example. Chains of length 2 Permutation: inversion and coinversion ¼=62453 inversion: {(i - j): i > j } coinversion: {(i - j): i < j } inv(¼)= 9 : { 62, 64, 6, 65, 63, 2, 4, 43, 53} coinv(¼)=6: { 24, 25, 23, 45, 5, 3 } where [k] p.q is the (p,q)-integer p k- +p k-2 q + + pq k-2 + q k-. 23
On words over { n, 2 n 2,, k n k } A word is an arrangement of n, 2 n 2,, k n k 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] 24
inv(¼) coinv(¼) Theorem [Kasraoui 0] The pair (ne2, se2) has a symmetric joint distribution over the set of 0-fillings of a moon polyomino with any given column sum. 25
various mixed statistics Bicolor the rows of M and mixed by the position of the top cell/ bottom cell top-mixed statistic (S,M): and bottom-mixed statistic (S,M): and 26
Mix by the charge of a corner cell Positive chains and Negative chains and 27
Symmetry on mixed statistics Theorem. [Chen, Wang, Y, Zhao 0; Wang &Y 3 ] 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. 28
The model is special enough! Many things happen inside rectangles! 29
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 0-fillings of Ferrers diagrams [de Mier 07] for multi-graphs with fixed degree sequences 30
It applies to other patterns. Both papers compared patterns J k and One can get more Wilf-equivalent pairs. 3
Applies to symmetric fillings [Bousquet-Melou, Steingrimsson 05] symmetric 0-fillings of symmetric Ferrers diagrams involution 32
Example 3. The major index For a word a a 2 a n, a descent is a position i such that a i > a i+. maj(w) = { i : i 2 DES(w) }. [MacMahon 96] The major index is equadistributed to inv(w) over words. 33
Example 3. The major index For a word a a 2 a n, a descent is a position i such that a i > a i+. maj(w) = { i : i 2 DES(w) }. [MacMahon 96] The major index is equadistributed to inv(w) over words. [Chen, Poznanovik, Y & Yang 0] The major index can be extended to 0-fillings of moon polyominoes, which has the same distribution as ne 2. 34
Foata s map Φ with inv(φ(w))=maj(w) Recursive Definition: If w has length, Φ(w)=w. Otherwise, w= w a, then Φ(w) = γ a (Φ (w )) a w =w w n- a v v n- a Φ γ a w w n- u u n- v v n- 35
Many transformations! [CPYY] Foata-type transformations can be defined on fillings of left-aligned stack polyominoes which carry maj to ne 2 36
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. ne 2 (F) = ne 2 (g(f)) 37
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 38
Relation to other areas Free probability- noncrossing diagrams 39
Crossings appear in the combinatorial interpretations of Mixed moments of random variables Moments of orthogonal polynomials Linearization coefficients 40
Graph optimization and layout: A partition of the edges into k-sets of noncrossing (non-nesting) edges 4
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 42
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Combinatorial computational biology: RNA pseudo knot structures 44
T H A N K Y O U V E R Y M U C H! 45