Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers

Abstract

The first problem addressed by this article is the enumeration of some families of patternavoiding inversion sequences. We solve some enumerative conjectures left open by the foundational work on the topics by Corteel et al., some of these being also solved independently by Lin, and Kim and Lin. The strength of our approach is its robustness: we enumerate four families F1 ⊂ F2 ⊂ F3 ⊂ F4 of pattern-avoiding inversion sequences ordered by inclusion using the same approach. More precisely, we provide a generating tree (with associated succession rule) for each family Fi which generalizes the one for the family Fi−1.The second topic of the paper is the enumeration of a fifth family F5 of pattern-avoiding inversion sequences (containing F4). This enumeration is also solved via a succession rule, which however does not generalize the one for F4. The associated enumeration sequence, which we call the powered Catalan numbers, is quite intriguing, and further investigated. We provide two different succession rules for it, denoted ΩpCat and Ω steady , and show that they define two types of families enumerated by powered Catalan numbers. Among such families, we introduce the steady paths, which are naturally associated with Ω steady . They allow us to bridge the gap between the two types of families enumerated by powered Catalan numbers: indeed, we provide a size-preserving bijection between steady paths and valley-marked Dyck paths (which are naturally associated with ΩpCat).Along the way, we provide several nice connections to families of permutations defined by the avoidance of vincular patterns, and some enumerative conjectures. arXiv:1808.04114v2 [math.CO] 14 Dec 2018 Proposition 2. Any inversion sequence e = (e 1 , . . . , e n ) is a Catalan inversion sequence if and only if for any i, with 1 ≤ i < n, if e i forms a weak descent, i.e. e i ≥ e i+1 , then e i < e j , for all j > i + 1.Proof. The forward direction is clear. The backwards direction can be proved by contrapositive. More precisely, suppose there are three indices i < j < k, such that e i ≥ e j , e k . Then, if e j = e i+1 , e i forms a weak descent and the fact that e i ≥ e k concludes the proof. Otherwise, since e i ≥ e j , there must be an index i , with i ≤ i < j, such that e i forms a weak descent and e i ≥ e k . This concludes the proof as well.

Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences

“…These examples indeed do not fit into our idea of what generalization/specialization of succession rules should be. Despite the restrictive character of our proposed definition, we believe that it applies to all our examples of the current paper, and of the other papers [5,9]. We leave open the questions whether the "correct" definition should be a bit less restrictive to allow for more instances to fit in, and whether it should be on the contrary more restrictive, to prevent other undesirable examples.…”

“…Indeed, it may be the case that the underlying growths for the classes A and B have nothing in common. This applies for instance to Dyck and Motzkin paths, with their growths presented in [4], or to families of pattern-avoiding inversion sequences (namely, avoiding the triple of relations (≥, −, ≥) and (≥, ≥, >), respectively) with their growths defined in [5].…”

“…Our first contribution is to provide a continuum from Catalan to Baxter via Schröder that is visible at the abstract level of succession rules. In this paper, as well as in our recent works [5,9], we consider several generating trees and their associated succession rules, and we focus on succession rules that generalize, or conversely specialize, well-known succession rules. Although this can be understood at a rather informal level (and this is actually how we originally worked), we propose a formalization of this idea of generalizing (resp.…”

Asymptotic normality of consecutive patterns in permutations encoded by generating trees with one‐dimensional labels

“…Generating trees have been used in the last 20 years to establish several enumerative results for various combinatorial classes of partitions, permutations, polyominoes, and many other objects (see for instance [3, 4, 8, 10, 12, 22, 24, 26, 50, 51, 54]). We refer to [5] and to the Ph.D. thesis of Guerrini [37, Chapter 1] for two interesting presentations of generating trees and associated enumeration techniques through generating functions.…”

Consecutive Patterns in Inversion Sequences

“…Their systematic enumeration of classical patterns of length 3 turns out to be interesting in its own right, as it connects these patterns in inversion sequences to well-known sequences such as Bell numbers, Euler up/down numbers, Fibonacci numbers and Schröder numbers. Since the appearance of the work of Corteel et al and Mansour and Shattuck, the interest in patterns in inversion sequences has been on the rise, and research in this area continues to proliferate [5,6,13,14,16,18,22]. Let us start with some basic definitions.…”