An involution on restricted Laguerre histories and its applications

TL;DR

This paper introduces a reflection-like involution on restricted Laguerre histories and demonstrates its applications in establishing equidistribution results and discovering new Mahonian permutation statistics.

Contribution

It presents a novel involution on restricted Laguerre histories and explores its applications in permutation statistics and combinatorial interpretations.

Findings

Discovered seven new Mahonian statistics.

Established new equidistribution results for permutation statistics.

Linked Yan-Zhou-Lin bijection with Kreweras complement.

Abstract

Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, and on the other hand, yield combinatorial interpretations for the moments of Laguerre polynomials via Flajolet's combinatorial theory of continued fractions. In this paper, we first introduce a reflection-like involution on restricted Laguerre histories. Then, we demonstrate its power by composing this involution with three bijections due to Fran\ccon-Viennot, Foata-Zeilberger, and Yan-Zhou-Lin, respectively. A host of equidistribution results involving various (multiset-valued) permutation statistics follow from these applications. As byproducts, seven apparently new Mahonian statistics present themselves; new interpretations of known Mahonian statistics are discovered as well. Finally, in our effort to show the interconnections between these Mahonian statistics, we are naturally led to a new link between the variant Yan-Zhou-Lin bijection and the Kreweras complement.