Simsun permutations, simsun successions and simsun patterns

TL;DR

This paper introduces new classes of permutations called simsun successions, studies their combinatorial properties, and connects them to existing mathematical objects like q-Eulerian polynomials and set partitions.

Contribution

It defines simsun successions and patterns, explores their statistical properties, and establishes novel bijections and interpretations in permutation and partition theory.

Findings

Characterization of simsun permutations avoiding pattern 321

Combinatorial interpretation of q-Eulerian polynomials

Bijection between simsun pattern 132 avoiding permutations and set partitions

Abstract

In this paper, we introduce the definitions of simsun succession, simsun cycle succession and simsun pattern. In particular, the ordinary simsun permutations are permutations avoiding simsun pattern 321. We study the descent and peak statistics on permutations avoiding simsun successions. We give a combinatorial interpretation of the q-Eulerian polynomials introduced by Brenti (J. Combin. Theory Ser. A 91 (2000), 137-170). We also present a bijection between permutations avoiding simsun pattern 132 and set partitions.