A new decomposition of ascent sequences and Euler–Stirling statistics

Abstract

As shown by Bousquet-Mélou-Claesson- Dukes-Kitaev (2010), ascent sequences can be used to encode (2 + 2)-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power seriesIn this paper, we present a novel way to recursively decompose ascent sequences, which leads to:• 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. • a far-reaching generalization of the generating function formula for (asc, zero) due toJelínek. 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. • 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 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.

Asymptotics and statistics on Fishburn matrices and their generalizations

“…These statistics are classified as Stirling statistics (or roughly statistics with logarithmic mean and variance); see [21]. On the other hand, the diagonal size that we address in this paper represents another Stirling statistic, which may also have interpretations in terms of other structures.…”

“…Fishburn matrices, introduced in the 1970s in the context of interval orders (in order theory) and directed graphs (see [1,18,23,41]), are nonnegative, upper-triangular ones without zero row or column. They have later found to be bijectively equivalent to several other combinatorial structures such as (2 + 2)-free posets, ascent sequences, pattern-avoiding permutations, patternavoiding inversion sequences, Stoimenow matchings, and regular chord diagrams; see, for instance, [6,14,21,30,35] and Section 2 for more information. In addition to their rich combinatorial connections, the corresponding asymptotic enumeration and the finer distributional properties are equally enriching and challenging, as we will explore in this paper.…”

“…From a combinatorial viewpoint, the Fishburn numbers also enumerate several seemingly unrelated structures some of which are listed as follows; see [6,9,14,13,16,18,21,30,33,35,43] for the bijective and algebraic proofs of these equinumerosity.…”

“…Stoimenow [39] found a recursive formula for the numbers of regular linearized chord diagram with a given length of the leftmost chord. Subsequently, it was discovered [6,21,30,34,35,43] that these numbers are equivalent to the following ones:…”

“…Substituting 1 + vz by Λ(vz) and 1 + z by Λ(z) proves (2.11). Alternatively, it is known that the generating function for the size of the first row (marked by v) is given by (see [21,33])…”