On Wilf Equivalence for Alternating Permutations

Abstract

In this paper, we obtain several new classes of Wilf-equivalent patterns for alternating permutations. In particular, we prove that for any nonempty pattern $\tau$, the patterns $12\ldots k\oplus\tau$ and $k\ldots 21\oplus\tau$ are Wilf-equivalent for alternating permutations, paralleling a result of Backelin, West, and Xin for Wilf equivalence for permutations.

(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.…”

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

“…Combining Theorem 1.2 and Lemma 3.17, we derive that A n (I k ⊕ τ ) = A n (J k ⊕ τ ) for any nonempty permutation τ and for any positive integer k which was first proved by Yan [36]. Similarly, by Theorem 1.5 and Lemma 3.17, we have AI n (I k ⊕ τ ) = AI n (J k ⊕ τ ) for any nonempty pattern τ and for any positive integer k, completing the proof of Theorem 1.8.…”

“…Note that Backelin-West-Xin [3] proved that S n (I k ⊕ τ ) = S n (J k ⊕ τ ) , which has been extended to involutions by Bousquet-Mélou and Steingrímsson [12] and to alternating permutations by Yan [36]. Hence, Theorem 1.8 can be viewed as a parallel work of the above results.…”

“…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]).…”