On the joint distributions of succession and Eulerian statistics

The motivation of this paper is to investigate the joint distribution of succession and Eulerian statistics. We first investigate the enumerators for the joint distribution of descents, big ascents and successions over all permutations in the symmetric group. In order to extend a result of Diaconis et al. (2014) [16], we show that two triple set-valued statistics of permutations are equidistributed. We then introduce the definition of proper left-to-right minimum, and discover that the joint distribution of the succession and proper left-to-right minimum statistics over permutations is a symmetric distribution. In the final part, we discuss the relationship between the fix and cyc $(p,q)$-Eulerian polynomials and the joint distribution of succession and Eulerian-type statistics. In particular, we give a concise derivation of the generating function for a six-variable Eulerian polynomial.