In this talk, I will first review some previous works on Fishburn numbers that are studied by Bousquet-Mélou, Bringmann, Claesson, Dukes, Fishburn, Kitaev, Jelínek, Levande, Li, Parviainen, Remmel, Rhoades, Steingrı́msson, Yan and Zagier. Next, I will present our main results on Fishburn numbers, ascent sequences and Euler--Stirling statistics. Our central contribution is the discovery of a new decomposition of ascent sequences, which leads to

1. a calculation of the Euler–Stirling distribution on ascent sequences, including the num-
bers 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;

2. a far-reaching generalization of a generating function formula for (asc,zero) due to Jelínek. 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;

3. 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 (2 − 1)-avoiding inversion sequences. This is achieved via a decomposition of (2 − 1)-avoiding inversion sequences parallel to that of ascent sequences.

This is joint work with Fu, Lin, Yan and Zhou. After that, if time permits, I will show our work-in-progress, together with Hsien-Kuei Hwang, on the asymptotics of Fishburn-like numbers and the limiting distributions of Euler--Stirling statistics over Fishburn objects.