A new decomposition of ascent sequences and Euler--Stirling statistics

As shown by Bousquet-Melou--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 Parviainens conjecture on the equidistribution of $zero$ and $max$. (ii) a far-reaching generalization of the generating function formula for $(asc,zero)$ due to Jelinek. 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.