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...
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.
We introduce new combinatorial structures, called fighting fish, that generalize directed convex polyominoes by allowing them to branch out of the plane into independent substructures. On the one hand the combinatorial structure of fighting fish appears to be particularly rich: we show that their generating function with respect to the perimeter and number of tails is algebraic, and we conjecture a mysterious multivariate equidistribution property with the left ternary trees introduced by Del Lungo et al On the other hand, fighting fish provide a simple and natural model of random branching surfaces which displays original features: in particular, we show that the average area of a uniform random fighting fish with perimeter 2n is of order n5/4: to the best of our knowledge this behaviour is non-standard and suggests that we have identified a new universality class of random structures.
The development of Next Generation Sequencing has had a major impact on the study of genetic sequences, and in particular, on the advancement of metagenomics, whose aim is to identify the microorganisms that are present in a sample collected directly from the environment. In this paper, we describe a new lightweight alignment-free and assembly-free framework for metagenomic classification that compares each unknown sequence in the sample to a collection of known genomes. We take advantage of the combinatorial properties of an extension of the Burrows-Wheeler transform, and we sequentially scan the required data structures, so that we can analyze unknown sequences of large collections using little internal memory. For the best of our knowledge, this is the first approach that is assembly-and alignment-free, and is not based on k-mers. We show that our experiments confirm the effectiveness of our approach and the high accuracy even in negative control samples. Indeed we only classify 1 short read on 5, 726, 358 random shuffle reads. Finally, the results are comparable with those achieved by read-mapping classifiers and by k-mer based classifiers.
We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the m-skinny slicings and the m-row-restricted slicings, for m ∈ N. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any m.
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