Unlabeled $(2+2)$-free posets, ascent sequences and pattern avoiding permutations

Abstract

International audience We present statistic-preserving bijections between four classes of combinatorial objects. Two of them, the class of unlabeled $(\textrm{2+2})$-free posets and a certain class of chord diagrams (or involutions), already appeared in the literature, but were apparently not known to be equinumerous. The third one is a new class of pattern avoiding permutations, and the fourth one consists of certain integer sequences called $\textit{ascent sequences}$. We also determine the generating function of these classes of objects, thus recovering a non-D-finite series obtained by Zagier for chord diagrams. Finally, we characterize the ascent sequences that correspond to permutations avoiding the barred pattern $3\bar{1}52\bar{4}$, and enumerate those permutations, thus settling a conjecture of Pudwell. Nous présentons des bijections, transportant de nombreuses statistiques, entre quatre classes d'objets. Deux d'entre elles, la classe des EPO (ensembles partiellement ordonnés) sans motif $(\textrm{2+2})$ et une certaine classe d'involutions, sont déjà apparues dans la littérature. La troisième est une classe de permutations à motifs exclus, et la quatrième une classe de suites que nous appelons $\textit{suites à montées}$. Nous déterminons ensuite la série génératrice de ces classes, retrouvant ainsi un résultat prouvé par Zagier pour les involutions sus-mentionnées. La série obtenue n'est pas D-finie. Apparemment, le fait qu'elle compte aussi les EPO sans motif $(\textrm{2+2})$ est nouveau. Finalement, nous caractérisons les suites à montées qui correspondent aux permutations évitant le motif barré $3\bar{1}52\bar{4}$ et énumérons ces permutations, ce qui démontre une conjecture de Pudwell.

Asymptotics and statistics on Fishburn matrices and their generalizations

“…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.…”

“…First, the Taylor coefficients of the inner product on the left-hand side of (1.1) alternating in sign, it is unclear if the coefficient of z n in the sum-of-product expression is positive for all positive n, much less its factorial growth order shown on the right-hand side. Second, since 6 π 2 < 1, the right-hand side of (1.1) is exponentially smaller than n!, which equals [z n ] 1 j n 1 − (1 − z) j . More precisely, we will prove that (see Lemma 12 and 14) max…”

“…where ρ = 12 eπ 2 = 2ρ, so there is indeed a heavy exponential cancellation involved in the sum on the left-hand side. Third, in addition to the connection to linearly independent Vassiliev invariants, the Fishburn numbers have now been known to enumerate many different combinatorial objects; see Section 2, OEIS sequence A022493 and [4,6,8,9,14,13,16,18,30,31,33,35,43,44] for more information. Finally, Zagier's proof of (1.1) relies crucially on the unusual pair of identities (1.2)…”

“…As a succinct representation tool for interval orders (see [23,18]), Fishburn matrices (called IOmatrices in [23], characteristic matrices in [17,18], and composition matrices in [12], and named so in [9]) offer not only algorithmic but also combinatorial advantages, and over the years their study was largely enriched by the corresponding developments in combinatorial enumeration and bijections, following notably Bousquet-Mélou et al's paper [6]. In particular, the useful database OEIS [29] played a key role in linking the various structures in different areas some of which will be briefly described later.…”

“…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.…”