Eulerian pairs and Eulerian recurrence systems

TL;DR

This paper explores a duality relation between Eulerian recurrences and systems, unifying various polynomial decompositions and deriving properties like unimodality and alternating increase in related enumerative polynomials.

Contribution

It introduces a generalized duality framework for Eulerian recurrences, unifying multiple polynomial decompositions and deriving new properties of associated enumerative polynomials.

Findings

Flag descent polynomials for hyperoctahedral group are characterized.

Both ascent-plateau and left ascent-plateau polynomials for Stirling permutations are unimodal.

The paper establishes a duality relation that unifies various polynomial decompositions.

Abstract

In this paper, we characterize a duality relation between Eulerian recurrences and Eulerian recurrence systems, which generalizes and unifies Hermite-Biehler decompositions of several enumerative polynomials, including flag descent polynomials for hyperoctahedral group, flag ascent-plateau polynomials for Stirling permutations, up-down run polynomials for symmetric group and alternating run polynomials for hyperoctahedral group. As applications, we derive some properties of associated enumerative polynomials. In particular, we find that both the ascent-plateau polynomials and left ascent-plateau polynomials for Stirling permutations are alternatingly increasing, and so they are unimodal with modes in the middle.