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Vincent Vatter

Publications and source records attributed to Vincent Vatter.

At least 19 recordsLinked to original sources
arXiv2026

Chains and unique transitive orientations of prime graphs

We give a short, conceptual proof that prime graphs have at most two transitive orientations, a much-quoted result of Gallai. Our proof uses chains, introduced by Chudnovsky, Kim, Oum, and Seymour, which provide a transparent characterization of primality. Transitivity induces a forcing relation on edges; using chains, we show that any two edges of a prime graph are equivalent under this relation, and thus any transitive orientation is unique up to reversal.

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arXiv2026

Permutation Wordle

We introduce a guessing game, ``Permutation Wordle,'' in which a guesser attempts to recover a setter's hidden permutation of the set $\{1, \ldots, n\}$. In each round, the guesser submits a word over the alphabet $\{1, \ldots, n\}$, and, as in the game Wordle, learns which entries are correct. We describe a natural strategy and prove that it is optimal in a strong sense: for every $r$, it solves at least as many secrets within $r$ rounds as any possible strategy. The number of permutations it solves in exactly $k+1$ rounds is the Eulerian number $A(n,k)$.

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arXiv2026

Log-concavity of subsequence counts of words

Given a word over a finite alphabet, consider the sequence that counts, for each length, the number of distinct subsequences of that length. In 1976, Chase proved that this sequence is log-concave. His proof uses a triangular array indexed by the prefixes of the word together with a meticulous analysis of ratios of several sums. We instead decompose according to the first letter, which reduces the proof to a weighted average.

math.CO↗
arXiv2026

Cyclomatic numbers and permutations

We show that several apparently different aspects of a permutation are all tied to a single quantity, the cyclomatic number of its inversion graph. Every reduced word for the permutation orders the edges of the inversion graph one at a time, with the edges from first-occurrence letters forming a spanning forest and the edges from repeated letters accounting for the rest; the number of repeated letters is therefore the cyclomatic number. The excess of permutation cycles over sum components is also at most this quantity, and it follows that the gap between the Coxeter and reflection lengths is at least the cyclomatic number and at most twice it. When the inversion graph is a forest, these results unify classical characterizations of the boolean permutations due to Edelman, to Tenner, and to Petersen and Tenner. We also give a new proof that every connected acyclic inversion graph is a caterpillar.

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arXiv2026

Boolean--Eulerian numbers

We study decreasing binary trees in which every vertex with two children is colored red or blue. We construct two bijections. The first, to ordered set partitions into odd-sized blocks each arranged as an alternating permutation, shows that the exponential generating function of these trees is $1/(1-\tan z)$. The second, to nonplane decreasing 1-2 trees paired with a binary label on each non-root vertex, proves combinatorially that the count equals $2^{n-1}$ times the~$n$th Euler number. Refining by the number of right edges yields the Boolean--Eulerian polynomials, which are an explicit algebraic transform of the classical Eulerian polynomials. The Foata--Strehl orbit decomposition, recast in the decreasing-binary-tree model, gives a direct combinatorial proof of gamma-positivity, and the algebraic transform carries real-rootedness and interlacing of zeros from the Eulerian polynomials to the Boolean--Eulerian polynomials.

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arXiv2026

Linear clique-width and modular decomposition

A hereditary class of graphs has bounded clique-width if and only if its prime members do, but this lifting property fails for linear clique-width. We prove that a hereditary class has bounded linear clique-width if and only if its prime members do and it contains neither all quasi-threshold graphs nor all complements of quasi-threshold graphs. This generalizes a result of Brignall, Korpelainen, and Vatter, who established the result for cographs.

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arXiv2023

Bounds on the lettericity of graphs

Lettericity measures the minimum size of an alphabet needed to represent a graph as a letter graph, where vertices are encoded by letters, and edges are determined by an underlying decoder. We prove that all graphs on~$n$ vertices have lettericity at most approximately $n - \tfrac{1}{2} \log_2 n$ and that almost all graphs on $n$ vertices have lettericity at least $n - (2 \log_2 n + 2 \log_2 \log_2 n)$.

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arXiv2022

Uncountably many enumerations of well-quasi-ordered permutation classes

We construct an uncountable family of well-quasi-ordered permutation classes, each with a distinct enumeration sequence. This disproves a conjecture that all well-quasi-ordered permutation classes have algebraic generating functions, and in fact shows that many such classes lack D-finite or D-algebraic generating functions. Our construction is based on an uncountably large collection of factor-closed, well-quasi-ordered binary languages due to Pouzet.

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arXiv2022

Three coloring via triangle counting

In the first partial result toward Steinberg's now-disproved three coloring conjecture, Abbott and Zhou used a counting argument to show that every planar graph without cycles of lengths 4 through 11 is 3-colorable. Implicit in their proof is a fact about plane graphs: in any plane graph of minimum degree 3, if no two triangles share an edge, then triangles make up strictly less than 2/3 of the faces. We show how this result, combined with Kostochka and Yancey's resolution of Ore's conjecture for k = 4, implies that every planar graph without cycles of lengths 4 through 8 is 3-colorable.

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arXiv2021

Letter graphs and geometric grid classes of permutations

We uncover a connection between two seemingly unrelated notions: lettericity, from structural graph theory, and geometric griddability, from the world of permutation patterns. Both of these notions capture important structural properties of their respective classes of objects. We prove that these notions are equivalent in the sense that a permutation class is geometrically griddable if and only if the corresponding class of inversion graphs has bounded lettericity.

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arXiv2021

Letter graphs and modular decomposition

We prove that if the prime graphs in a graph class have bounded lettericity, then the entire class has bounded lettericity if and only if it does not contain arbitrary large matchings, co-matchings, or a family of graphs that we call stacked paths.

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arXiv2021

Labelled well-quasi-order for permutation classes

While the theory of labelled well-quasi-order has received significant attention in the graph setting, it has not yet been considered in the context of permutation patterns. We initiate this study here, and show how labelled well quasi order provides a lens through which to view and extend previous well-quasi-order results in the permutation patterns literature. Connections to the graph setting are emphasised throughout. In particular, we establish that a permutation class is labelled well-quasi-ordered if and only if its corresponding graph class is also labelled well-quasi-ordered.

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arXiv2020

How many pop-stacks does it take to sort a permutation?

Pop-stacks are variants of stacks that were introduced by Avis and Newborn in 1981. Coincidentally, a 1982 result of Unger implies that every permutation of length n can be sorted by n-1 passes through a deterministic pop-stack. We give a new proof of this result inspired by Knuth's zero-one principle.

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arXiv2020

Bijective proofs of proper coloring theorems

The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynomial case and the bijection for broken circuits is just a restriction of the one for all spanning subgraphs.

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arXiv2018

Containing all permutations

Numerous versions of the question "what is the shortest object containing all permutations of a given length?" have been asked over the past fifty years: by Karp (via Knuth) in 1972; by Chung, Diaconis, and Graham in 1992; by Ashlock and Tillotson in 1993; and by Arratia in 1999. The large variety of questions of this form, which have previously been considered in isolation, stands in stark contrast to the dearth of answers. We survey and synthesize these questions and their partial answers, introduce infinitely more related questions, and then establish an improved upper bound for one of these questions.

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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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