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.
Highlights
Introduction and resultWe let Sn denote the set of all permutations of [n] := {1, . . . , n}
We investigate the enumeration of some Motzkin objects, namely the sets In(4321), In(3412), and Sn(321, 3 ̄142), according to the numbers of crossings and nestings
The bijection of Biane maps an involution into a Motzkin path whose down steps are labelled with an integer that does not exceed their height, while the other steps are unlabelled
Summary
Introduction and resultWe let Sn denote the set of all permutations of [n] := {1, . . . , n}. We define some statistics over Motzkin paths and we prove a preliminary result that is fundamental for the rest of the paper (see Theorem 5). By the same way, when we decompose the area statistic according to up and horizontal steps, we obtain the following proposition.
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