Semi-Baxter and Strong-Baxter: Two Relatives of the Baxter Sequence

Abstract

In this paper, we enumerate two families of pattern-avoiding permutations: those avoiding the vincular pattern 2 41 3, which we call semi-Baxter permutations, and those avoiding the vincular patterns 2 41 3, 3 14 2 and 3 41 2, which we call strong-Baxter permutations. We call semi-Baxter numbers and strong-Baxter numbers the associated enumeration sequences. We prove that the semi-Baxter numbers enumerate in addition plane permutations (avoiding 2 14 3). The problem of counting these permutations was open and has given rise to several conjectures, which we also prove in this paper.For each family (that of semi-Baxter -or equivalently, plane -and that of strong-Baxter permutations), we describe a generating tree, which translates into a functional equation for the generating function. For semi-Baxter permutations, it is solved using (a variant of) the kernel method: this gives an expression for the generating function while also proving its D-finiteness. From the obtained generating function, we derive closed formulas for the semi-Baxter numbers, a recurrence that they satisfy, as well as their asymptotic behavior. For strong-Baxter permutations, we show that their generating function is (a slight modification of) that of a family of walks in the quarter plane, which is known to be non D-finite. arXiv:1702.04529v3 [math.CO] 11 Jan 2018⊆ Av(2 41 3, 3 41 2) ⊆ Figure 1: Sequences from Catalan to factorial numbers, with nested families of pattern-avoiding permutations that they enumerate.The focus of this paper is the study of the two sequences of semi-Baxter and strong-Baxter numbers.We deal with the semi-Baxter sequence (enumerating semi-Baxter permutations) in Section 3. It has been proved in [22] (as a special case of a general statement) that this sequence also enumerates plane permutations, defined by the avoidance of 2 14 3. This sequence is referenced as A117106 in [23]. We first give a more specific proof that plane permutations and semi-Baxter permutations are equinumerous, by providing a common generating tree (or succession rule) with two labels for these two families. Basics and references about generating trees can be found in Section 2.We solve completely the problem of enumerating semi-Baxter permutations (or equivalently, plane permutations), pushing further the techniques that were used to enumerate Baxter permutations in [10]. Namely, we start from the functional equation associated with our succession rule for semi-Baxter permutations, and we solve it using variants of the kernel method [10,21]. This results in an expression for the generating function for semi-Baxter permutations, showing that this generating function is D-finite 2 . From it, we obtain several formulas for the semi-Baxter numbers: first, a complicated closed formula; second, a simple recursive formula; and third, three simple closed formulas that were conjectured by D. Bevan [7].The problem of enumerating plane permutations was posed by M. Bousquet-Mélou and S. Butler in [12]. Some conjectures related to this enumeration problem w...

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.…”

“…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.…”

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

“…As proved in [11,Section 3], once Proposition 14 has been established, the enumeration of the family I(≥, >, −) is obtained by applying the obstinate kernel method as discussed for the family I(≥, ≥, ≥).…”

“…In this paper we study five families of inversion sequences which form a hierarchy for the inclusion order. The enumeration of these classes -by well-known sequences, such as those of the Catalan, the Baxter, and the newly introduced semi-Baxter numbers [11] -was originally conjectured in the first version of [26]. These conjectures have attracted the attention of a fair number of combinatorialists, resulting in proofs for all of them, independently of our paper.…”

“…This family of inversion sequences was originally conjectured in [26] to be counted by the sequence A117106 [27]. The validity of this conjecture follows from [11] where the numbers of sequence A117106 are named semi-Baxter, hence the name semi-Baxter inversion sequences. We recall the first terms of this enumeration sequence 1, 2, 6, 23, 104, 530, 2958, 17734, 112657, 750726, 5207910, 37387881, 276467208, .…”