Proof of a bi-symmetric septuple equidistribution on ascent sequences

Abstract

It is well known since the seminal work by Bousquet-Mélou, 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 bi-vincular 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 area bijective proof of a bi-symmetric septuple equidistribution of Euler-Stirling statistics on ascent sequences, involving the number of ascents (\(\mathsf{asc}\)), the number of repeated entries (\(\mathsf{rep}\)), the number of zeros (\(\mathsf{zero}\)), the number of maximal entries (\(\mathsf{max}\)), the number of right-to-left minima (\(\mathsf{rmin}\)) and two auxiliary statistics; 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 \((\mathsf{asc},\mathsf{rep},\mathsf{zero},\mathsf{max})\) and \((\mathsf{rep},\mathsf{asc},\mathsf{max},\mathsf{zero})\) 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). Mathematics Subject Classifications: 05A15, 05A19Keywords: Ascent sequences, equidistributions, Euler-Stirling statistics, Fishburn numbers, basic hypergeometric series