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

TL;DR

This paper studies the distribution of crossings and nestings in specific Motzkin objects and related permutations, deriving continued fractions, recursion formulas, and introducing new q-Motzkin numbers.

Contribution

It introduces new q-Motzkin numbers and provides explicit distributions of crossings and nestings over certain pattern-avoiding involutions and permutations.

Findings

Distribution formulas for crossings and nestings

Continued fraction representations and recursion formulas

Introduction of two new q-Motzkin numbers

Abstract

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.