Counting Permutations by Simsun Successions

Abstract

In this paper, we introduce the definitions of simsun succession statistics and simsun patterns. In addition to its original definition by Brenti, we give two more combinatorial interpretations of the $q$-Eulerian polynomials using simsun successions. We also present a bijection between permutations avoiding the simsun pattern $132$ and set partitions.