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

As shown by Bousquet-M'elou–Claesson–Dukes–Kitaev (2010), ascent sequences can be used to encode $(\mathbf{2}+\mathbf{2})$-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power series $$\sum _{m=1}^{\mathrm{\infty }}\prod _{i=1}^m(1-(1-t{)}^i)\mathrm{.}$$ 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 ($\text{asc}$), repeated entries $(\text{rep})$, zeros ($\text{zero}$) and maximal entries ($\max$). In particular, this confirms and extends Dukes and Parviainen’s conjecture on the equidistribution of $\text{zero}$ and $\max$. (ii) a far-reaching generalization of the generating function formula for $(\text{asc},\text{zero})$ due to Jel'inek. This is accomplished via a bijective proof of the quadruple equidistribution of $(\text{asc},\text{rep},\text{zero},\max )$ and $(\text{rep},\text{asc},\mathrm{i}n,\text{zero})$, where $\mathrm{i}n$ denotes the right-to-left minima statistic of ascent sequences. (iii) an extension of a conjecture posed by Levande, which asserts that the pair $(\text{asc},\text{zero})$ on ascent sequences has the same distribution as the pair $(\text{rep},\max )$ on $(\mathbf{2}-\mathbf{1})$-avoiding inversion sequences. This is achieved via a decomposition of $(\mathbf{2}-\mathbf{1})$-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 $(\text{asc},\text{rep},\text{zero},\max )$ and $(\text{rep},\text{asc},\max ,\text{zero})$ are equidistributed on ascent sequences.