148 research outputs found

    Generating trees and pattern avoidance in alternating permutations

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    We extend earlier work of the same author to enumerate alternating permutations avoiding the permutation pattern 2143. We use a generating tree approach to construct a recursive bijection between the set A_{2n}(2143) of alternating permutations of length 2n avoiding 2143 and standard Young tableaux of shape (n, n, n) and between the set A_{2n + 1}(2143) of alternating permutations of length 2n + 1 avoiding 2143 and shifted standard Young tableaux of shape (n + 2, n + 1, n). We also give a number of conjectures and open questions on pattern avoidance in alternating permutations and generalizations thereof.Comment: 21 pages. To be presented at FPSAC 2010. Comments welcome

    Pattern avoidance for alternating permutations and reading words of tableaux

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 67-69).We consider a variety of questions related to pattern avoidance in alternating permutations and generalizations thereof. We give bijective enumerations of alternating permutations avoiding patterns of length 3 and 4, of permutations that are the reading words of a "thickened staircase" shape (or equivalently of permutations with descent set {k, 2k, 3k, . . .}) avoiding a monotone pattern, and of the reading words of Young tableaux of any skew shape avoiding any of the patterns 132, 213, 312, or 231. Our bijections include a simple bijection involving binary trees, variations on the Robinson-Schensted-Knuth correspondence, and recursive bijections established via isomorphisms of generating trees.by Joel Brewster Lewis.Ph.D

    Pattern avoiding alternating involutions

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    We enumerate and characterize some classes of alternating and reverse alternating involutions avoiding a single pattern of length three or four. If on one hand the case of patterns of length three is trivial, on the other hand, the length four case is more challenging and involves sequences of combinatorial interest, such as Motzkin and Fibonacci numbers

    On a refinement of Wilf-equivalence for permutations

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    Recently, Dokos et al. conjectured that for all k,m≥1k, m\geq 1, the patterns 12…k(k+m+1)…(k+2)(k+1) 12\ldots k(k+m+1)\ldots (k+2)(k+1) and (m+1)(m+2)…(k+m+1)m…21(m+1)(m+2)\ldots (k+m+1)m\ldots 21 are majmaj-Wilf-equivalent. In this paper, we confirm this conjecture for all k≥1k\geq 1 and m=1m=1. In fact, we construct a descent set preserving bijection between 12…k(k−1) 12\ldots k (k-1) -avoiding permutations and 23…k123\ldots k1-avoiding permutations for all k≥3k\geq 3. As a corollary, our bijection enables us to settle a conjecture of Gowravaram and Jagadeesan concerning the Wilf-equivalence for permutations with given descent sets
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