Crossings and nestings over some Motzkin objects and $q$-Motzkin numbers

Abstract

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.