Staircases, dominoes, and the growth rate of 1324-avoiders

Bevan, David; Brignall, Robert; Price, Andrew Elvey; Pantone, Jay

doi:10.1016/j.endm.2017.06.029

Cited by 8 publications

(15 citation statements)

References 15 publications

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“…For example, suppose H is the valid hook configuration of 21123214567 shown in Figure 6. Referring to the induced coloring shown in Figure 7, we find that (9,4), (10,5)…”

Section: Trees and Valid Hook Configurationsmentioning

confidence: 98%

Stack-sorting for Words

Defant,

Kravitz

2018

Preprint

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We introduce operators hare and tortoise, which act on words as natural generalizations of West's stack-sorting map. We show that the heuristically slower algorithm tortoise can sort words arbitrarily faster than its counterpart hare. We then generalize the combinatorial objects known as valid hook configurations in order to find a method for computing the number of preimages of any word under these two operators. We relate the question of determining which words are sortable by hare and tortoise to more classical problems in pattern avoidance, and we derive a recurrence for the number of words with a fixed number of copies of each letter (permutations of a multiset) that are sortable by each map. In particular, we use generating trees to prove that the -uniform words on the alphabet [n] that avoid the patterns 231 and 221 are counted by the ( + 1)-Catalan number 1 n+1 ( +1)n n. We conclude with several open problems and conjectures.

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“…For example, suppose H is the valid hook configuration of 21123214567 shown in Figure 6. Referring to the induced coloring shown in Figure 7, we find that (9,4), (10,5)…”

Section: Trees and Valid Hook Configurationsmentioning

confidence: 98%

Stack-sorting for Words

Defant,

Kravitz

2018

Preprint

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“…The generating function for the last symmetry class, containing Av(1324) and Av(4231), remains unknown. Through about a dozen articles [8,25,28,29,36,37,64,66,67,95,117], the first fifty terms of the counting sequence of Av(1324) are known explicitly and rough bounds on the exponential growth are known.…”

Section: Enumeration Of Permutation Classesmentioning

confidence: 99%

Combinatorial Exploration: An algorithmic framework for enumeration

Albert¹,

Bean²,

Claesson³

et al. 2022

Preprint

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Combinatorial Exploration is a new domain-agnostic algorithmic framework to automatically and rigorously study the structure of combinatorial objects and derive their counting sequences and generating functions. We describe how it works and provide an open-source Python implementation. As a prerequisite, we build up a new theoretical foundation for combinatorial decomposition strategies and combinatorial specifications.We then apply Combinatorial Exploration to the domain of permutation patterns, to great effect. We rederive hundreds of results in the literature in a uniform manner and prove many new ones. These results can be found in a new public database, the Permutation Pattern Avoidance Library (PermPAL) at https://permpal.com. Finally, we give three additional proofs-of-concept, showing examples of how Combinatorial Exploration can prove results in the domains of alternating sign matrices, polyominoes, and set partitions.

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“…For example, if π ∈ S n , then leg(π) = n+1 if and only if π avoids 231. The legal spaces of 145326 are (0, 1), (1,2), (4,5), (5,6), (6,7), so leg(145326) = 5. Imagine adding a new point somewhere to the left of all points in the plot of a permutation π.…”

Section: -Stack-sortable Permutationsmentioning

confidence: 99%

“…The articles [31] and [35] give different proofs that new combinatorial objects called "fighting fish" are counted by the numbers W 2 (n). The authors of [2] studied what they called "n-point dominoes," and they have found that there are W 2 (n + 1) such objects.…”

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confidence: 99%

Counting 3-stack-sortable permutations

Defant

2020

Journal of Combinatorial Theory, Series A

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We prove a "decomposition lemma" that allows us to count preimages of certain sets of permutations under West's stack-sorting map s. As a first application, we give a new proof of Zeilberger's formula for the number W2(n) of 2-stack-sortable permutations in Sn. Our proof generalizes, allowing us to find an algebraic equation satisfied by the generating function that counts 2-stack-sortable permutations according to length, number of descents, and number of peaks. This is also the first proof of this formula that generalizes to the setting of 3-stack-sortable permutations. Indeed, the same method allows us to obtain a recurrence relation for W3(n), the number of 3-stacksortable permutations in Sn. Hence, we obtain the first polynomial-time algorithm for computing these numbers. We compute W3(n) for n ≤ 174, vastly extending the 13 terms of this sequence that were known before. We also prove the first nontrivial lower bound for lim n→∞ W3(n) 1/n , showing that it is at least 8.659702. Invoking a result of Kremer, we also prove that lim n→∞ Wt(n) 1/n ≥ ( √ t + 1) 2 for all t ≥ 1, which we use to improve a result of Smith concerning a variant of the stack-sorting procedure. Our computations allow us to disprove a conjecture of Bóna, although we do not yet know for sure which one.In fact, we can refine our methods to obtain a recurrence for W3(n, k, p), the number of 3-stacksortable permutations in Sn with k descents and p peaks. This allows us to gain a large amount of evidence supporting a real-rootedness conjecture of Bóna. Using part of the theory of valid hook configurations, we give a new proof of a γ-nonnegativity result of Brändén, which in turn implies an older result of Bóna. We then answer a question of the current author by producing a set A ⊆ S11 such that σ∈s −1 (A) x des(σ) has nonreal roots. We interpret this as partial evidence against the same real-rootedness conjecture of Bóna that we found evidence supporting. Examining the parities of the numbers W3(n), we obtain strong evidence against yet another conjecture of Bóna. We end with some conjectures of our own.

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Staircases, dominoes, and the growth rate of 1324-avoiders

Cited by 8 publications

References 15 publications

Stack-sorting for Words

Stack-sorting for Words

Combinatorial Exploration: An algorithmic framework for enumeration

Counting 3-stack-sortable permutations

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