Ascent序列的双对称七元等分布的证明
Proof of a bi-symmetric septuple equidistribution on ascent sequences
摘要：
众所周知，自Bousquet-Mélou、Claesson、Dukes和Kitaev（2010）的开创性工作以来，关于几个自然统计量的上升序列的某些精化是双射的，其对应的精化是避免双偶模式的无$({\bf2+2})$偏序集和置换。不同的乘加细枚举的上升序列和其他双射等价结构随后被广泛研究的各种作者。在本文中，我们的主要贡献是1。对上升序列统计量的双对称七元等分布进行了双射证明，包括上升数（asc）、重复项数（rep）、零数（zero）、最大项数（max）、从右到左最小值数（rmin）和两个辅助统计量; 2.给出了在基$q$中展开为解析函数的无终止基本超几何$_4ϕ_3$级数在$q=1$周围的一个新的变换公式，并利用它证明了上升序列上的两个（双）对称四重等分布.我们发现的一个副产品包括确认关于上升序列上的欧拉-斯特林统计量四元组（asc，rep，zero，max）和（rep，asc，max，zero）之间的双对称等分布的猜想，这是由Foata（1977）的双欧拉等分布引起的，最近由Fu，Lin，Yan，Zhou和第一作者（2018）提出。
英文摘要：
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 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 $_4ϕ_3$ 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).
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