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

Beaton, Nicholas R.; Bouvel, Mathilde; Guerrini, Veronica; Rinaldi, Simone

doi:10.37236/7375

Cited by 5 publications

(9 citation statements)

References 16 publications

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“…The marked shapes are used to define the embedding of a CBT into an object. This is done by recursively embedding triples 6 For example, for CBTs it is (by design) trivial to perform the embedding. From the product definition we get the embedding:…”

Section: Geometrical Familiesmentioning

confidence: 99%

“…F 13 : Catalan floor plans [6] Catalan floor plans are equivalence classes of rectangles containing rectangles. A size n floor plan rectangle contains n non-nested rectangles -called the 'rooms' of the plan.…”

Section: Appendixmentioning

confidence: 99%

“…A history of Catalan numbers by I. Pak can be found in the appendix of [4] and a very extensive biography of Catalan (and Bell numbers ) was compiled by H. Gould [5]. New Catalan families are regularly being discovered, for example the Catalan floor plans [6].…”

Section: Introductionmentioning

confidence: 99%

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A Universal Bijection for Catalan Structures

Brak¹

2018

Preprint

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Free magmas can be defined using the standard universal mapping principle for free structures. We provide an alternative characterisation of free magmas which is simpler to use when proving some combinatorial structure is a free magma: the magma has to have a norm and unique factorisation. This characterisation can be used to show conjectured Catalan families are (once a suitable product is defined) free magmas. The norm (a certain 'size' function) partitions the magma base set into subsets enumerated by generalised Catalan numbers. The universal isomorphism between free magmas gives a universal bijection -essentially one bijection algorithm for any pair of Catalan families. The morphism property ensures the bijection is recursive. The universal bijection allows us to give some rigour to the idea of an "embedding" bijection between Catalan objects which, in many cases, show how to embed an element of one Catalan family into one of a different family. Multiplication on the right (respectively left) by the free magma generator gives rise to the right (respectively left) Narayana statistic. We discuss the relation between the symbolic method for Catalan families and the magma structure on the base set defined by the symbolic method. This shows which "atomic" element is the magma generator. The appendix gives the magma structure for 14 Catalan families.

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Section: Geometrical Familiesmentioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

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A Universal Bijection for Catalan Structures

Brak¹

2018

Preprint

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“…Moreover, the ways in which these growths are encoded by labels in succession rules are also each a natural extension of the case of the immediately smaller family. Hence, these examples provide another illustration of the idea of generalizing/specializing succession rules that we discussed in details in [6,Section 2.2]. The outcome of the discussion in [6, Section 2.2] is the following proposed definition for generalization/specialization of succession rules.…”

Section: Content Of the Papermentioning

confidence: 99%

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

Beaton¹,

Bouvel²,

Guerrini³

et al. 2019

Theoretical Computer Science

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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.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Borga

2021

Random Struct Algorithms

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We consider uniform random permutations drawn from a family enumerated through generating trees. We develop a new general technique to establish a central limit theorem for the number of consecutive occurrences of a fixed pattern in such permutations. We propose a technique to sample uniform permutations in such families as conditioned random colored walks. Building on that, we derive the behavior of the consecutive patterns in random permutations studying properties of the consecutive increments in the corresponding random walks. The method applies to families of permutations with a one‐dimensional‐labeled generating tree (together with some technical assumptions) and implies local convergence for random permutations in such families. We exhibit ten different families of permutations, most of them being permutation classes, that satisfy our assumptions. To the best of our knowledge, this is the first work where generating trees—which were introduced to enumerate combinatorial objects—have been used to establish probabilistic results.

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

Cited by 5 publications

References 16 publications

A Universal Bijection for Catalan Structures

A Universal Bijection for Catalan Structures

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

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

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