The ascent-plateau statistics on Stirling permutations

TL;DR

This paper introduces new ascent-plateau variants for Stirling permutations, analyzes their properties using grammatical and group action methods, and establishes equidistribution results for certain bistatistics.

Contribution

It presents novel ascent-plateau variants, a unified polynomial refinement, and proves bistatistics are equidistributed using advanced combinatorial techniques.

Findings

Introduction of flag ascent-plateau, double ascent, and descent-plateau statistics.

Unified refinement of ascent and ascent-plateau polynomials.

Proof of equidistribution of bistatistics via group actions.

Abstract

In this paper, several variants of the ascent-plateau statistic are introduced, including flag ascent-plateau, double ascent and descent-plateau. We first study the flag ascent-plateau statistic on Stirling permutations by using context-free grammars. We then present a unified refinement of the ascent polynomials and the ascent-plateau polynomials. In particular, by using Foata and Strehl's group action, we prove two bistatistics over the set of Stirling permutations of order n are equidistributed.