Profile Classes and Partial Well-Order for Permutations
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TLDR
In this article, it was shown that the profile class of permutations is partially well-ordered if and only if a related graph is a forest and the antichain of the graph is not a permutation.Abstract:
It is known that the set of permutations, under the pattern containment ordering, is not a partial well-order. Characterizing the partially well-ordered closed sets (equivalently: down sets or ideals) in this poset remains a wide-open problem. Given a $0/\pm1$ matrix $M$, we define a closed set of permutations called the profile class of $M$. These sets are generalizations of sets considered by Atkinson, Murphy, and Ruskuc. We show that the profile class of $M$ is partially well-ordered if and only if a related graph is a forest. Related to the antichains we construct to prove one of the directions of this result, we construct exotic fundamental antichains, which lack the periodicity exhibited by all previously known fundamental antichains of permutations.read more
Citations
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Grid Classes and the Fibonacci Dichotomy for Restricted Permutations
Sophie Huczynska,Vincent Vatter +1 more
TL;DR: In this article, the authors introduced and characterised grid classes, which are natural generalisations of other well-studied permutation classes, and gave a new, short proof of the Fibonacci dichotomy: the number of permutations of length $n$ in a permutation class is either at least as large as the first or polynomial.
Journal ArticleDOI
Geometric grid classes of permutations
TL;DR: Using a mixture of geometric and language theoretic methods, the authors proved that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions.
Journal ArticleDOI
Small permutation classes
TL;DR: In this paper, the authors established a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number approximately 2.20557, for which there are only countably many permutations classes of growth rate (Stanley-Wilf limit) less than 1.
Posted Content
Geometric grid classes of permutations
TL;DR: Using a mixture of geometric and language theoretic methods, this paper proved that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions.
Posted Content
Permutation classes
TL;DR: A survey on permutation classes for the upcoming book Handbook of Enumerative Combinatorics can be found in this paper, where the authors present a survey of permutation algorithms.
References
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Book
Combinatorics on words
TL;DR: Perrin and Perrin this article showed that square free words and idempotent semigroups can be expressed in terms of free monoids, and the critical factorization theorem of Van der Waerden's theorem.
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Combinatorics on Words
TL;DR: Words (strings of symbols) are fundamental in computer processing, and nearly all computer software use algorithms on strings.
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Restricted permutations
TL;DR: The partial order on permutations that underlies the idea of restriction and which gives rise to sets of sequences closed under taking subsequences is studied.
Journal ArticleDOI
Sorting Using Networks of Queues and Stacks
TL;DR: The model is established that a switchyard is an acyclic directed graph, with a unique source and a unique sink, and each vertex represents a siding, which has the property of having indefinite storage space.