No abstract

Let st = {st 1 , st 2 , . . . , st k } be a set of k statistics on permutations with k ≥ 1. We say that two given subset of permutations T and T ′ are st-Wilf-equivalent if the joint distributions of all statistics in st over the sets of T-avoiding permutations S n (T) and T ′ -avoiding permutations S n (T ′ ) are the same. The main purpose of this paper is the (cr,nes)-Wilf-equivalence classes for all single pattern in S 3 , where cr and nes denote respectively the statistics number of crossings and nestings. One of the main tools that we use is the bijection Θ : S n (321) → S n (132) which was originally exhibited by Elizalde and Pak in [10]. They proved that the bijection Θ preserves the number of fixed points and excedances. Since the given formulation of Θ is not direct, we show that it can be defined directly by a recursive formula. Then, we prove that it also preserves the number of crossings. Due to the fact that the sets of non-nesting permutations and 321-avoiding permutations are the same, these properties of the bijection Θ leads to an unexpected result related to the q,p-Catalan numbers of Randrianarivony defined in [17].

We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of $4321$-avoiding involutions, the set of $3412$-avoiding involutions, and the set of $(321,3\bar{1}42)$-avoiding permutations. To get our results, we exploit the bijection of Biane restricted to the sets of $4321$- and $3412$-avoiding involutions which was characterized by Barnabei et al. and the bijection between $(321,3\bar{1}42)$-avoiding permutations and Motzkin paths, presented by Chen et al.. Furthermore, we manipulate the obtained continued fractions to get the recursion formulas for the polynomial distributions of crossings and nestings, and it follows that the results involve two new $q$-Motzkin numbers.

In this paper, we investigate on enumeration of some Motzkin objects according to the numbers of crossings and nestings. More precisely, we compute and express in terms of continued fractions the distributions of the statistics numbers of crossings and nestings over the sets of 4321-and 3412-involutions and the set of (321, 3 142)-avoiding permutations. To get our results, we exploit the bijection of Biane restricted to the sets of 4321 and 3412-avoiding involutions which was characterized by Barnabei et al. and the bijection Chen et al. between (321, 3 142)-avoiding permutations and Motzkin paths. Furthermore, we manipulate the obtained continued fractions to get the recursion formulas for the polynomial distributions of crossings and nestings and we discover that results involve two new q-Motzkin numbers.