The inversion number statistic for inversion sequences

TL;DR

This paper introduces the inversion number statistic for inversion sequences, connecting it to classical permutation statistics and combinatorial numbers, and explores its distribution through a $q$-analog approach.

Contribution

It extends the inversion number statistic to inversion sequences and links it to well-known combinatorial distributions and numbers, unifying various classical results.

Findings

Recovered classical permutation statistics like Stirling, Mahonian, and Eulerian distributions.

Connected inversion number statistics to Catalan and Narayana numbers.

Included the enumeration of involutions in the symmetric group as a special case.

Abstract

Inversion sequences, also known as subexcedant sequences, form a fundamental class of objects in enumerative combinatorics. In this paper, we study the joint distribution of five statistics on inversion sequences. While several statistics on inversion sequences have been extensively investigated, our contribution is to introduce the inversion number statistic, originally defined for permutations, into the context of inversion sequences. As special cases, we recover classical permutation statistics, including the Stirling, Mahonian and Eulerian distributions, as well as the Catalan and Narayana numbers. Somewhat unexpectedly, our specializations also include the number of involutions in the symmetric group. Our study arises from a $q$-analog of Comtet's expansion formula obtained by substituting the classical derivative operator $D$ with the $q$-derivative operator $D_q$.