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Vincent Vatter

Publications and source records attributed to Vincent Vatter.

At least 37 records · Page 2Linked to original sources
arXiv2019

Growth rates of permutation classes: categorization up to the uncountability threshold

In the antecedent paper to this it was established that there is an algebraic number $ξ\approx 2.30522$ such that while there are uncountably many growth rates of permutation classes arbitrarily close to $ξ$, there are only countably many less than $ξ$. Here we provide a complete characterization of the growth rates less than $ξ$. In particular, this classification establishes that $ξ$ is the least accumulation point from above of growth rates and that all growth rates less than or equal to $ξ$ are achieved by finitely based classes. A significant part of this classification is achieved via a reconstruction result for sum indecomposable permutations. We conclude by refuting a suggestion of Klazar, showing that $ξ$ is an accumulation point from above of growth rates of finitely based permutation classes.

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arXiv2019

Growth rates of permutation classes: from countable to uncountable

We establish that there is an algebraic number $ξ\approx 2.30522$ such that while there are uncountably many growth rates of permutation classes arbitrarily close to $ξ,$ there are only countably many less than $ξ$. Central to the proof are various structural notions regarding generalized grid classes and a new property of permutation classes called concentration. The classification of growth rates up to $ξ$ is completed in a subsequent paper.

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arXiv2019

Rationality for subclasses of 321-avoiding permutations

We prove that every proper subclass of the 321-avoiding permutations that is defined either by only finitely many additional restrictions or is well quasi-ordered has a rational generating function. To do so we show that any such class is in bijective correspondence with a regular language. The proof makes significant use of formal languages and of a host of encodings, including a new mapping called the panel encoding that maps languages over the infinite alphabet of positive integers avoiding certain subwords to languages over finite alphabets.

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arXiv2018

A counterexample regarding labelled well-quasi-ordering

Korpelainen, Lozin, and Razgon conjectured that a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by only finitely many minimal forbidden induced subgraphs is labelled well-quasi-ordered, a notion stronger than that of $n$-well-quasi-order introduced by Pouzet in the 1970s. We present a counterexample to this conjecture. In fact, we exhibit a hereditary property of graphs which is well-quasi-ordered by the induced subgraph order and defined by finitely many minimal forbidden induced subgraphs yet is not $2$-well-quasi-ordered. This counterexample is based on the widdershins spiral, which has received some study in the area of permutation patterns.

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arXiv2018

Generating Permutations with Restricted Containers

We investigate a generalization of stacks that we call $\mathcal{C}$-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that $\mathcal{C}$-machines generate, and how these systems of functional equations can frequently be solved by either the kernel method or, much more easily, by guessing and checking. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by $\mathcal{C}$-machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their enumerations, seem to not have D-finite generating functions.

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arXiv2017

Universal layered permutations

We establish an exact formula for the length of the shortest permutation containing all layered permutations of length $n$, proving a conjecture of Gray.

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arXiv2017

An elementary proof of Bevan's theorem on the growth of grid classes of permutations

Bevan established that the growth rate of a monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. We give an elementary and self-contained proof of a generalization of this result using only Stirling's Formula, the method of Lagrange multipliers, and the singular value decomposition of matrices.

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arXiv2017

On the dimension of downsets of integer partitions and compositions

We characterize the downsets of integer partitions (ordered by containment of Ferrers diagrams) and compositions (ordered by the generalized subword order) which have finite dimension in the sense of Dushnik and Miller. In the case of partitions, while the set of all partitions has infinite dimension, we show that every proper downset of partitions has finite dimension. For compositions we identify four minimal downsets of infinite dimension and establish that every downset which does not contain one of these four has finite dimension.

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arXiv2016

The Complexity of Pattern Matching for $321$-Avoiding and Skew-Merged Permutations

The Permutation Pattern Matching problem, asking whether a pattern permutation $π$ is contained in a permutation $τ$, is known to be NP-complete. In this paper we present two polynomial time algorithms for special cases. The first algorithm is applicable if both $π$ and $τ$ are $321$-avoiding; the second is applicable if $π$ and $τ$ are skew-merged. Both algorithms have a runtime of $O(kn)$, where $k$ is the length of $π$ and $n$ the length of $τ$.

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arXiv2016

On the growth of merges and staircases of permutation classes

There is a well-known upper bound on the growth rate of the merge of two permutation classes. Curiously, there is no known merge for which this bound is not achieved. Using staircases of permutation classes, we provide sufficient conditions for this upper bound to be achieved. In particular, our results apply to all merges of principal permutation classes. We end by demonstrating how our techniques can be used to reprove a result of Bóna.

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arXiv2016

Small permutation classes

We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number $κ$, approximately 2.20557, for which there are only countably many permutation classes of growth rate (Stanley-Wilf limit) less than $κ$ but uncountably many permutation classes of growth rate $κ$, answering a question of Klazar. We go on to completely characterize the possible sub-$κ$ growth rates of permutation classes, answering a question of Kaiser and Klazar. Central to our proofs are the concepts of generalized grid classes (introduced herein), partial well-order, and atomicity (also known as the joint embedding property).

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arXiv2016

An Erdős--Hajnal analogue for permutation classes

Let $\mathcal{C}$ be a permutation class that does not contain all layered permutations or all colayered permutations. We prove that there is a constant $c$ such that every permutation in $\mathcal{C}$ of length $n$ contains a monotone subsequence of length $cn$.

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arXiv2016

Linear Clique-Width for Hereditary Classes of Cographs

The class of cographs is known to have unbounded linear clique-width. We prove that a hereditary class of cographs has bounded linear clique-width if and only if it does not contain all quasi-threshold graphs or their complements. The proof borrows ideas from the enumeration of permutation classes.

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arXiv2015

Well-Quasi-Order for Permutation Graphs Omitting a Path and a Clique

We consider well-quasi-order for classes of permutation graphs which omit both a path and a clique. Our principle result is that the class of permutation graphs omitting $P_5$ and a clique of any size is well-quasi-ordered. This is proved by giving a structural decomposition of the corresponding permutations. We also exhibit three infinite antichains to show that the classes of permutation graphs omitting $\{P_6,K_6\}$, $\{P_7,K_5\}$, and $\{P_8,K_4\}$ are not well-quasi-ordered.

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arXiv2015

Permutation classes

This is a survey on permutation classes for the upcoming book Handbook of Enumerative Combinatorics.

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arXiv2014

Pattern-Avoiding Involutions: Exact and Asymptotic Enumeration

We consider the enumeration of pattern-avoiding involutions, focusing in particular on sets defined by avoiding a single pattern of length 4. As we demonstrate, the numerical data for these problems demonstrates some surprising behavior. This strange behavior even provides some very unexpected data related to the number of 1324-avoiding permutations.

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arXiv2014

On the Rearrangement Conjecture for Generalized Factor Order Over $\mathbb{P}$

The Rearrangement Conjecture states that if two words over $\mathbb{P}$ are Wilf-equivalent in the factor order on $\mathbb{P}^\ast$ then they are rearrangements of each other. We introduce the notion of strong Wilf-equivalence and prove that if two words over $\mathbb{P}$ are strongly Wilf-equivalent then they are rearrangements of each other. We further conjecture that Wilf-equivalence implies strong Wilf-equivalence.

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