Eulerian pairs and Eulerian recurrence systems

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.

Stability of combinatorial polynomials and its applications

“…Remark 5.17. The alternatingly increasing property of M n,2 (x) and M n,2 (x) was also proved in [54,Theorem 12] in a different way. Here our Proposition 5.16 gives a stronger result than the alternatingly increasing property.…”

“…In this subsection, we will consider the peak polynomials on the generalization of r-Stirling permutations, which extend the dual set of 2-Stirling permutations in [54]. Let π = π 1 π 2 .…”