Enumerative Combinatorics with Fillings of Polyominoes
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1 Enumerative Combinatorics with Fillings of Polyominoes Catherine Yan Texas A&M University Tulane, March 205
2 Tulane, March 205 2
3 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 Tulane, March 205 3
4 Part I: Symmetry of the longest chains Permutations: (increasing subsequence) (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. Tulane, March 205 4
5 Crossings and nestings in matchings of [2n] (cr 2, ne 2 ) are symmetric! e.g. cr 2 ne # noncrossing matchings of [2n] = # nonnesting matchings of [2n] = nth Catalan number Tulane, March 205 5
6 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 Tulane, March 205 6
7 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 Tulane, March 205 7
8 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]. #{M: cr(m)=i, ne(m)=j } = #{ M: cr(m)=j, ne(m)= i } Corollary. # matchings with no k-crossing = # matchings with no k-nesting Tulane, March 205 8
9 Idea: Oscillating tableau: a sequence of Ferrers diagrams ;=λ 0, λ,, λ 2n =; s.t. λ i = λ i- +/ - ; ; Tulane, March 205 9
10 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. (cr, ne) ß à max row/column lengths. Theorem [CDDSY] Taking conjugation in the tableaux exchanges cr(m) and ne(m). Tulane, March 205 0
11 Set Partitions of [n] A graphical representation π= {,4, 5, 7} {2,6} {3} Theorem. [CDDSY] (cr(¼), ne(¼)) has a symmetric distribution over all partitions of [n]. Tulane, March 205
12 Filling of the triangular board Crossing: anti-identity submatrix (NE-chain) Nesting: identity submatrix (SE-chain) Tulane, March 205 2
13 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 Tulane, March 205 3
14 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 restricted 0-fillings of F with n s.. Tulane, March 205 4
15 Generalized triangulation of n-gon k-triangulation: no k+ diagonals that are mutually intersecting Tulane, March 205 5
16 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. Tulane, March 205 6
17 Catalan number implies symmetry! try to avoid F( max s, no k-ne ) = F(max s, no k-se) Tulane, March 205 7
18 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. Tulane, March 205 8
19 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) Tulane, March 205 9
20 The General Model: fillings of moon polyominoes Polyomino: a finite set of square cells Moon polyomino: Convex intersection-free (no skew shape) Tulane, March
21 Fillings of moon polyominoes Assign an integer to each square Permuta -tions Words Matchings Set partitions Graphs Ferrers diagram Stack polyomino Moon polyomino Tulane, March 205 2
22 Part II: Combinatorics of fillings of moon polyominoes Northeast and southeast chains Forbidden patterns Transformations Connections to other objects Tulane, March
23 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-. Tulane, March
24 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] Tulane, March
25 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. Tulane, March
26 various mixed statistics Bicolor the rows of M and mix the 2- chains by the position of the top cell/ bottom cell top-mixed statistic (S,M): and bottom-mixed statistic (S,M): and Tulane, March
27 Mix by the charge of a corner cell Positive chains and Negative chains and Tulane, March
28 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. Tulane, March
29 The model is special enough! Many things happen inside rectangles! Tulane, March
30 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 Tulane, March
31 It applies to other patterns. Both papers compared patterns J k and One can get more Wilf-equivalent pairs. Tulane, March 205 3
32 Applies to symmetric fillings [Bousquet-Melou, Steingrimsson 05] symmetric 0-fillings of symmetric Ferrers diagrams involution [Bloom, Saracino 2] explain the connection between algebraic and combinatorial approaches by modifying the growth diagram algorithm Tulane, March
33 The model is flexible One can change the fillings, or One can change the polyomino. Tulane, March
34 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. Tulane, March
35 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. Tulane, March
36 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- Tulane, March
37 Many transformations! [CPYY] Foata-type transformations can be defined on fillings of left-aligned stack polyominoes which carry maj to ne 2 Tulane, March
38 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)) Tulane, March
39 And more Lattice path counting and descents in Ferrers diagrams Rook placement with restrictions Pattern avoidance and appearances Simplicial complexes/schubert polynomials Tulane, March
40 Relation to other areas Free probability- noncrossing diagrams Tulane, March
41 Crossings appear in the combinatorial interpretations of Mixed moments of random variables Moments of orthogonal polynomials Linearization coefficients Tulane, March 205 4
42 Graph Optimization K-stack layout and k-queue layout: A partition of the edges into k-sets of noncrossing (non-nesting) edges Tulane, March
43 Stack- and Queue- numbers 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 Tulane, March
44 Tulane, March
45 Combinatorial computational biology: RNA pseudo knot structures Tulane, March
46 T H A N K Y O U V E R Y M U C H! Tulane, March
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