On pattern avoiding alternating permutations

Abstract

An alternating permutation of length n is a permutation π = π 1 π 2 · · · π n such that π 1 < π 2 > π 3 < π 4 > · · · . Let A n denote set of alternating permutations of {1, 2, . . . , n}, and let A n (σ) be set of alternating permutations in A n that avoid a pattern σ. Recently, Lewis used generating trees to enumerate A 2n (1234), A 2n (2143) and A 2n+1 (2143), and he posed several conjectures on the Wilf-equivalence of alternating permutations avoiding certain patterns. Some of these conjectures have been proved by Bóna, Xu and Yan. In this paper, we prove the two relations |A 2n+1 (1243)| = |A 2n+1 (2143)| and |A 2n (4312)| = |A 2n (1234)| as conjectured by Lewis.

(q,t)-Catalan numbers: gamma expansions, pattern avoidances, and the (−1)-phenomenon

“…Motivated by Lewis' work [25][26][27][28], many authors [5,10,33,42,43] have studied pattern avoidance on alternating permutations, especially the Wilf-equivalence problem for patterns of length four. As for alternating permutations that avoid two patterns of length four simultaneously, our results in section 6 concerning S n (2413, 3142) and S n (1342, 2431) appear to be new.…”

“…The statistics,,(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) and are constant on any orbit under the MFS-action.…”

“…The statistics,,(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) and (31-2) are constant on any orbit under the MFS-action.Proof. For π ∈ S n , when π(0) = π(n + 1) = n + 1, the cases (2-31) and (13-2) were proved by Brändén[7, Theorem 5.1].…”

On Wilf Equivalence for Alternating Permutations

“…In his paper [10], Lewis posed several conjectures on the enumeration of alternating permutations avoiding a given pattern of length 4 and 5. Some of these conjectures were proved by Bóna [5], Chen et al [6] and Xu et al [14].…”

International Comparison on the Application of Information Technology in High School Mathematics Textbooks

“…For example, various results have been obtained for pattern avoiding alternating permutations (see e.g. [10,15,21,22,23,24,28,35,36]) and pattern avoiding involutions (see e.g. [4,11,12,17,18,19]).…”