Proof of a bi-symmetric septuple equidistribution on ascent sequences

TL;DR

This paper proves a complex symmetric distribution of multiple statistics on ascent sequences using bijective methods and hypergeometric series, confirming a conjecture related to Euler--Stirling statistics.

Contribution

It introduces a bijective proof of a septuple equidistribution and a new hypergeometric transformation formula, advancing the understanding of ascent sequence statistics.

Findings

Established a bijective proof of a bi-symmetric septuple equidistribution.

Derived a new transformation formula for basic hypergeometric series.

Confirmed a conjecture on bi-symmetric quadruple equidistribution of Euler--Stirling statistics.

Abstract

It is well known since the seminal work by Bousquet-M\'elou, Claesson, Dukes and Kitaev (2010) that certain refinements of the ascent sequences with respect to several natural statistics are in bijection with corresponding refinements of $({\bf2+2})$-free posets and permutations that avoid a bivincular pattern. Different multiply-refined enumerations of ascent sequences and other bijectively equivalent structures have subsequently been extensively studied by various authors. In this paper, our main contributions are 1. a bijective proof of a bi-symmetric septuple equidistribution of statistics on ascent sequences, involving the number of ascents (asc), the number of repeated entries (rep), the number of zeros (zero), the number of maximal entries (max), the number of right-to-left minima (rmin) and two auxiliary statistics; 2. a new transformation formula for non-terminating basic hypergeometric $_4\phi_3$ series expanded as an analytic function in base $q$ around $q=1$, which is utilized to prove two (bi)-symmetric quadruple equidistributions on ascent sequences. A by-product of our findings includes the affirmation of a conjecture about the bi-symmetric equidistribution between the quadruples of Euler--Stirling statistics (asc,rep,zero,max) and (rep,asc,max,zero) on ascent sequences, that was motivated by a double Eulerian equidistribution due to Foata (1977) and recently proposed by Fu, Lin, Yan, Zhou and the first author (2018).