Emma Yu Jin

Two skew diagrams are defined to be equivalent if their corresponding skew Schur functions are equal. The equivalence classes for ribbons have been classified by Billera, Thomas and van Willigenburg in 2006. In this paper, we provide a complete characterization of equivalence classes for connected skew diagrams with exactly one $2\times m$ or $m\times 2$ block of boxes for all $m\ge 2$. In particular, possible sizes of equivalence classes are one, two or four, confirming special cases of the elusive conjecture on equivalent skew connected diagrams proposed by McNamara and van Willigenburg in 2009.

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

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).

As shown by Bousquet-M\'elou--Claesson--Dukes--Kitaev (2010), ascent sequences can be used to encode $({\bf2+2})$-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power series $$\sum_{m=1}^{\infty}\prod_{i=1}^m (1-(1-t)^i).$$ In this paper, we present a novel way to recursively decompose ascent sequences, which leads to: (i) a calculation of the Euler--Stirling distribution on ascent sequences, including the numbers of ascents ($\asc$), repeated entries $(\rep)$, zeros ($\zero$) and maximal entries ($\max$). In particular, this confirms and extends Dukes and Parviainen's conjecture on the equidistribution of $\zero$ and $\max$. (ii) a far-reaching generalization of the generating function formula for $(\asc,\zero)$ due to Jel\'inek. This is accomplished via a bijective proof of the quadruple equidistribution of $(\asc,\rep,\zero,\max)$ and $(\rep,\asc,\rmin,\zero)$, where $\rmin$ denotes the right-to-left minima statistic of ascent sequences. (iii) an extension of a conjecture posed by Levande, which asserts that the pair $(\asc,\zero)$ on ascent sequences has the same distribution as the pair $(\rep,\max)$ on $({\bf2-1})$-avoiding inversion sequences. This is achieved via a decomposition of $({\bf2-1})$-avoiding inversion sequences parallel to that of ascent sequences. This work is motivated by a double Eulerian equidistribution of Foata (1977) and a tempting bi-symmetry conjecture, which asserts that the quadruples $(\asc,\rep,\zero,\max)$ and $(\rep,\asc,\max,\zero)$ are equidistributed on ascent sequences.