arXiv subjects

Guillaume Chapuy

Publications and source records attributed to Guillaume Chapuy.

At least 37 records · Page 2Linked to original sources

arXiv2017

From Aztec diamonds to pyramids: steep tilings

We introduce a family of domino tilings that includes tilings of the Aztec diamond and pyramid partitions as special cases. These tilings live in a strip of $\mathbb{Z}^2$ of the form $1 \leq x-y \leq 2\ell$ for some integer $\ell \geq 1$, and are parametrized by a binary word $w\in\{+,-\}^{2\ell}$ that encodes some periodicity conditions at infinity. Aztec diamond and pyramid partitions correspond respectively to $w=(+-)^\ell$ and to the limit case $w=+^\infty-^\infty$. For each word $w$ and for different types of boundary conditions, we obtain a nice product formula for the generating function of the associated tilings with respect to the number of flips, that admits a natural multivariate generalization. The main tools are a bijective correspondence with sequences of interlaced partitions and the vertex operator formalism (which we slightly extend in order to handle Littlewood-type identities). In probabilistic terms our tilings map to Schur processes of different types (standard, Pfaffian and periodic). We also introduce a more general model that interpolates between domino tilings and plane partitions.

math.CO↗

arXiv2017

Laplacian matrices and spanning trees of tree graphs

If $G$ is a strongly connected finite directed graph, the set $\mathcal{T}G$ of rooted directed spanning trees of $G$ is naturally equipped with a structure of directed graph: there is a directed edge from any spanning tree to any other obtained by adding an outgoing edge at its root vertex and deleting the outgoing edge of the endpoint. Any Schrödinger operator on $G$, for example the Laplacian, can be lifted canonically to $\mathcal{T}G$. We show that the determinant of such a lifted Schrödinger operator admits a remarkable factorization into a product of determinants of the restrictions of Schrödinger operators on subgraphs of $G$ and we give a combinatorial description of the multiplicities using an exploration procedure of the graph. A similar factorization can be obtained from earlier ideas of C. Athaniasadis, but this leads to a different expression of the multiplicities, as signed sums on which the nonnegativity is not appearent. We also provide a description of the block structure associated with this factorization. As a simple illustration we reprove a formula of Bernardi enumerating spanning forests of the hypercube, that is closely related to the graph of spanning trees of a bouquet. Several combinatorial questions are left open, such as giving a bijective interpretation of the results.

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arXiv2016

Local convergence and stability of tight bridge-addable graph classes

A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if $\mathcal{G}$ is bridge-addable and $G_n$ is a uniform $n$-vertex graph from $\mathcal{G}$, then $G_n$ is connected with probability at least $(1+o_n(1))e^{-1/2}$. The constant $e^{-1/2}$ is best possible since it is reached for the class of all forests. In this paper we prove a form of uniqueness in this statement: if $\mathcal{G}$ is a bridge-addable class and the random graph $G_n$ is connected with probability close to $e^{-1/2}$, then $G_n$ is asymptotically close to a uniform $n$-vertex random forest in some local sense. For example, if the probability converges to $e^{-1/2}$, then $G_n$ converges in the sense of Benjamini-Schramm to the uniform infinite random forest $F_\infty$. This result is reminiscent of so-called "stability results" in extremal graph theory, with the difference that here the stable extremum is not a graph but a graph class.

math.CO↗

arXiv2016

A bijection for rooted maps on general surfaces

We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, that is orientable and non-orientable as well. This general construction requires new ideas and is more delicate than the special orientable case, but it carries the same information. In particular, it leads to a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable rooted connected maps of Euler characteristic $2-2h$, and of the algebraicity of their generating functions, similar to the one previously obtained in the orientable case via the Marcus-Schaeffer bijection. It also shows that the renormalization factor $n^{1/4}$ for distances between vertices is universal for maps on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation on any fixed surface converge in distribution when the size $n$ tends to infinity. Finally, we extend the Miermont and Ambjørn-Budd bijections to the general setting of all surfaces. Our construction opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.

math.CO↗

arXiv2015

Dimers on Rail Yard Graphs

We introduce a general model of dimer coverings of certain plane bipartite graphs, which we call rail yard graphs (RYG). The transfer matrices used to compute the partition function are shown to be isomorphic to certain operators arising in the so-called boson-fermion correspondence. This allows to reformulate the RYG dimer model as a Schur process, i.e. as a random sequence of integer partitions subject to some interlacing conditions. Beyond the computation of the partition function, we provide an explicit expression for all correlation functions or, equivalently, for the inverse Kasteleyn matrix of the RYG dimer model. This expression, which is amenable to asymptotic analysis, follows from an exact combinatorial description of the operators localizing dimers in the transfer-matrix formalism, and then a suitable application of Wick's theorem. Plane partitions, domino tilings of the Aztec diamond, pyramid partitions, and steep tilings arise as particular cases of the RYG dimer model. For the Aztec diamond, we provide new derivations of the edge-probability generating function, of the biased creation rate, of the inverse Kasteleyn matrix and of the arctic circle theorem.

math-ph↗

arXiv2015

Packing triangles in weighted graphs

Tuza conjectured that for every graph $G$, the maximum size $ν$ of a set of edge-disjoint triangles and minimum size $τ$ of a set of edges meeting all triangles satisfy $τ\leq 2ν$. We consider an edge-weighted version of this conjecture, which amounts to packing and covering triangles in multigraphs. Several known results about the original problem are shown to be true in this context, and some are improved. In particular, we answer a question of Krivelevich who proved that $τ\leq 2ν^*$ (where $ν^*$ is the fractional version of $ν$), and asked if this is tight. We prove that $τ\leq 2ν^*-\frac{1}{\sqrt{6}}\sqrt{ν^*}$ and show that this bound is essentially best possible.

math.CO↗

arXiv2015

Generating functions of bipartite maps on orientable surfaces

We compute, for each genus $g\geq 0$, the generating function $L_g\equiv L_g(t;p_1,p_2,\dots)$ of (labelled) bipartite maps on the orientable surface of genus $g$, with control on all face degrees. We exhibit an explicit change of variables such that for each $g$, $L_g$ is a rational function in the new variables, computable by an explicit recursion on the genus. The same holds for the generating function $F_g$ of rooted bipartite maps. The form of the result is strikingly similar to the Goulden/Jackson/Vakil and Goulden/Guay-Paquet/Novak formulas for the generating functions of classical and monotone Hurwitz numbers respectively, which suggests stronger links between these models. Our result complements recent results of Kazarian and Zograf, who studied the case where the number of faces is bounded, in the equivalent formalism of dessins d'enfants. Our proofs borrow some ideas from Eynard's "topological recursion" that he applied in particular to even-faced maps (unconventionally called "bipartite maps" in his work). However, the present paper requires no previous knowledge of this topic and comes with elementary (complex-analysis-free) proofs written in the perspective of formal power series.

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arXiv2014

The asymptotic number of $12..d$-Avoiding Words with $r$ occurrences of each letter $1,2, ..., n$

Following Ekhad and Zeilberger (The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, Dec 5 2014; see also arXiv:1412.2035), we study the asymptotics for large $n$ of the number $A_{d,r}(n)$ of words of length $rn$ having $r$ letters $i$ for $i=1..n$, and having no increasing subsequence of length $d$. We prove an asymptotic formula conjectured by these authors, and we give explicitly the multiplicative constant appearing in the result, answering a question they asked. These two results should make the OEIS richer by 100+25=125 dollars. In the case $r=1$ we recover Regev's result for permutations. Our proof goes as follows: expressing $A_{d,r}(n)$ as a sum over tableaux via the RSK correspondence, we show that the only tableaux contributing to the sum are "almost" rectangular (in the scale $\sqrt{n}$). This relies on asymptotic estimates for the Kotska numbers $K_{λ,r^n}$ when $λ$ has a fixed number of parts. Contrarily to the case $r=1$ where these numbers are given by the hook-length formula, we don't have closed form expressions here, so to get our asymptotic estimates we rely on more delicate computations, via the Jacobi-Trudi identity and saddle-point estimates.

math.CO↗

arXiv2014

Counting factorizations of Coxeter elements into products of reflections

In this paper, we count factorizations of Coxeter elements in well-generated complex reflection groups into products of reflections. We obtain a simple product formula for the exponential generating function of such factorizations, which is expressed uniformly in terms of natural parameters of the group. In the case of factorizations of minimal length, we recover a formula due to P. Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex case. For the symmetric group, our formula specializes to a formula of D. M. Jackson.

math.CO↗

arXiv2014

Simple recurrence formulas to count maps on orientable surfaces

We establish a simple recurrence formula for the number $Q_g^n$ of rooted orientable maps counted by edges and genus. We also give a weighted variant for the generating polynomial $Q_g^n(x)$ where $x$ is a parameter taking the number of faces of the map into account, or equivalently a simple recurrence formula for the refined numbers $M_g^{i,j}$ that count maps by genus, vertices, and faces. These formulas give by far the fastest known way of computing these numbers, or the fixed-genus generating functions, especially for large $g$. In the very particular case of one-face maps, we recover the Harer-Zagier recurrence formula. Our main formula is a consequence of the KP equation for the generating function of bipartite maps, coupled with a Tutte equation, and it was apparently unnoticed before. It is similar in look to the one discovered by Goulden and Jackson for triangulations, and indeed our method to go from the KP equation to the recurrence formula can be seen as a combinatorial simplification of Goulden and Jackson's approach (together with one additional combinatorial trick). All these formulas have a very combinatorial flavour, but finding a bijective interpretation is currently unsolved.

math.CO↗

arXiv2014

On the diameter of random planar graphs

We show that the diameter D(G_n) of a random labelled connected planar graph with n vertices is equal to n^{1/4+o(1)}, in probability. More precisely there exists a constant c>0 such that the probability that D(G_n) lies in the interval (n^{1/4-ε},n^{1/4+ε}) is greater than 1-\exp(-n^{cε}) for ε small enough and n>n_0(ε). We prove similar statements for 2-connected and 3-connected planar graphs and maps.

math.CO↗

arXiv2013

The local limit of unicellular maps in high genus

We show that the local limit of unicellular maps whose genus is proportional to the number of edges is a supercritical geometric Galton-Watson tree conditioned to survive. The proof relies on enumeration results obtained via the recent bijection given by the second author together with Feray and Fusy.

math.PR↗

arXiv2013

A simple model of trees for unicellular maps

We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the "recursive part" of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure. All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a bijective proof for each of them, in a unified way. For some of these formulas, this is the first bijective proof, e.g. the Harer-Zagier recurrence formula, the Lehman-Walsh formula and the Goupil-Schaeffer formula. We also discuss several applications of our construction: we obtain a new proof of an identity related to covered maps due to Bernardi and the first author, and thanks to previous work of the second author, we give a new expression for Stanley character polynomials, which evaluate irreducible characters of the symmetric group. Finally, we show that our techniques apply partially to unicellular 3-constellations and to related objects that we call quasi-constellations.

math.CO↗

arXiv2013

The representation of the symmetric group on m-Tamari intervals

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^{m}, which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S_n on these spaces is conjectured to be closely related to the natural representation of S_n on (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S_n acts on labelled intervals of T_n^{m} by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S_n. In particular, the dimension of the representation, that is, the number of labelled m-Tamari intervals of size n, is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables.

math.CO↗

arXiv2012

The vertical profile of embedded trees

Consider a rooted binary tree with n nodes. Assign with the root the abscissa 0, and with the left (resp. right) child of a node of abscissa i the abscissa i-1 (resp. i+1). We prove that the number of binary trees of size n having exactly n_i nodes at abscissa i, for l =< i =< r (with n = sum_i n_i), is $$ \frac{n_0}{n_l n_r} {{n_{-1}+n_1} \choose {n_0-1}} \prod_{l\le i\le r \atop i\not = 0}{{n_{i-1}+n_{i+1}-1} \choose {n_i-1}}, $$ with n_{l-1}=n_{r+1}=0. The sequence (n_l, ..., n_{-1};n_0, ..., n_r) is called the vertical profile of the tree. The vertical profile of a uniform random tree of size n is known to converge, in a certain sense and after normalization, to a random mesure called the integrated superbrownian excursion, which motivates our interest in the profile. We prove similar looking formulas for other families of trees whose nodes are embedded in Z. We also refine these formulas by taking into account the number of nodes at abscissa j whose parent lies at abscissa i, and/or the number of vertices at abscissa i having a prescribed number of children at abscissa j, for all i and j. Our proofs are bijective.

math.CO↗

arXiv2012

A note on the diameter of transportation polytopes with prescribed source degrees

Brightwell, van den Heuvel and Stougie proved that the diameter of an $m \times n$ transportation polytope is at most $8(m+n-2)$, a factor of eight away from the Hirsch Conjecture. This bound was improved to $3(m+n-1)$ by Hurkens. We investigate diameters for certain classes of transportation polytopes. Note: After the completion of this note, we discovered that the class of transportation polytopes studied in this note was already considered in Michel L. Balinski. On two special classes of transportation polytopes. Math. Programming Stud., 1:43-58, 1974. Michel L. Balinski and Fred J. Rispoli. Signature classes of transportation polytopes. Mathematical Programming, 60(2, Ser. A):127-144, 1993. These papers contain both refinements of our results and generalizations to more general classes of transportation problems. In view of these papers, this note will not be submitted for publication.

math.CO↗

arXiv2012

Tamari lattices and parking functions: proof of a conjecture of F. Bergeron

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^(m), which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of coinvariant spaces. He conjectured several intriguing formulas dealing with the enumeration of intervals in this lattice. One of them states that the number of intervals in T_n^(m) is $$ \frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$ This conjecture was proved recently, but in a non-bijective way, while its form strongly suggests a connection with plane trees. Here, we prove another conjecture of Bergeron, which deals with the number of labelled, intervals. An interval [P,Q] of T_n^(m) is labelled, if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. We prove that the number of labelled intervals in T_n^(m) is $$ {(m+1)^n(mn+1)^{n-2}}. $$ The form of these numbers suggests a connection with parking functions, but our proof is non-bijective. It is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. This equation involves a derivative and a divided difference, taken with respect to two additional variables. Solving this equation is the hardest part of the paper. Finding a bijective proof remains an open problem.

math.CO↗

arXiv2012

A note on a Cayley graph of S_n

Recently in graph theory several authors have studied the spectrum of the Cayley graph of the symmetric group S_n generated by the transpositions (1, i) for 2 <= i <= n. Several conjectures were made and partial results were obtained. The purpose of this note is to point out that, as mentioned also by P. Renteln, this problem is actually already solved in another context. Indeed it is equivalent to studying the spectrum of so-called Jucys-Murphy elements in the algebra of the symmetric group, which is well understood. The aforementioned conjectures are direct consequences of the existing theory. We also present a related result from P. Biane, giving an asymptotic description of this spectrum. We insist on the fact that this note does not contain any new results, but has only been written to convey the information from the algebraic combinatorics community to graph theorists.

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