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Stack-Sorting, Set Partitions, and Lassalle's Sequence
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TLDR
In this paper, a bijection between combinatorial objects known as valid hook configurations and certain weighted set partitions is presented, where permutations that have exactly one preimage under the (West) stack-sorting map are considered.Abstract:
We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new combinatorial interpretations of Lassalle's sequence. One of these interpretations involves permutations that have exactly one preimage under the (West) stack-sorting map. We prove that the sequences obtained by counting these permutations according to their first entries are symmetric, and we conjecture that they are log-concave. We also obtain new recurrence relations involving Lassalle's sequence and the sequence that enumerates valid hook configurations. We end with several suggestions for future work.read more
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Counting 3-stack-sortable permutations
TL;DR: A "decomposition lemma" that allows us to count preimages of certain sets of permutations under West's stack-sorting map $s$ is proved, and a new proof of Zeilberger's formula for the number of 2-stack-sortable permutations in $S_n$.
Journal ArticleDOI
Catalan intervals and uniquely sorted permutations
TL;DR: For each positive integer k, this paper considered five well-studied posets defined on the set of Dyck paths of semilength k and proved that uniquely sorted permutations avoiding various patterns are equinumerous with intervals in these posets.
Posted Content
Troupes, Cumulants, and Stack-Sorting
TL;DR: In this paper, it was shown that several sequences of free cumulants that count binary plane trees correspond to sequences of classical cumulant that count the decreasing versions of the same trees, and this surprising phenomenon holds for families of trees that we call troupes.
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Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations
TL;DR: In this paper, the authors enumerate classes of uniquely sorted permutations that avoid a pattern of length three and/or four by establishing bijections between these classes and various lattice paths.
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Stack-Sorting Preimages of Permutation Classes
TL;DR: The current paper represents a new approach, based on the theory of valid hook configurations, for solving classical enumerative problems, and provides several new combinatorial interpretations and identities involving known sequences, which paves the way for several new enumeratives problems.
References
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The Art in Computer Programming
Andrew Hunt,Dave Thomas +1 more
TL;DR: Here the authors haven’t even started the project yet, and already they’re forced to answer many questions: what will this thing be named, what directory will it be in, what type of module is it, how should it be compiled, and so on.
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Enumerative Combinatorics
TL;DR: This review of 3 Enumerative Combinatorics, by Charalambos A.good, does not support this; the label ‘Example’ is given in a rather small font followed by a ‘PROOF,’ and the body of an example is nonitalic, utterly unlike other statements accompanied by demonstrations.
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Lectures on the Combinatorics of Free Probability
Alexandru Nica,Roland Speicher +1 more
TL;DR: In this article, the authors present a case study of non-normal distribution and non-commutative joint distributions and define a set of basic combinatorics, such as non-crossing partitions, sum-of-free random variables, and products of free random variables.
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Combinatorics of Permutations
None Daiva Zadeike,Miklós Bóna +1 more
TL;DR: This book discusses Permutations as Genome Rearrangements, algorithms and permutations, and the proof of the Stanley-Wilf Conjecture.