June 2016

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115 Reads

Published by Centre pour la Communication Scientifique Directe (CCSD)

Online ISSN: 1365-8050

June 2016

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115 Reads

An {\it additive labeling} of a graph $G$ is a function $ \ell :V(G)
\o\mathbb{N}$, such that for every two adjacent vertices $ v $ and $ u$ of $ G
$, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $ ($ x \sim y $ means
that $ x $ is joined to $y$). An {\it additive number} of $ G $, denoted by
$\eta(G)$, is the minimum number $k $ such that $ G $ has a additive labeling $
\ell :V(G) \to \lbrace 1,...,k\rbrace $. An {\it additive choosability number}
of a graph $G$, denoted by $\eta_{\ell}(G) $, is the smallest number $k $ such
that $G$ has an additive labeling from any assignment of lists of size $k$ to
the vertices of $G$.
Seamone (2012) \cite{a80} conjectured that for every graph $G$, $\eta(G)=
\eta_{\ell}(G)$. We give a negative answer to this conjecture and we show that
for every $k$ there is a graph $G$ such that $ \eta_{\ell}(G)- \eta(G) \geq k$.
A {\it $(0,1)$-additive labeling} of a graph $G$ is a function $ \ell :V(G)
\rightarrow\{0,1\}$, such that for every two adjacent vertices $ v $ and $ u$
of $ G $, $ \sum_{w \sim v}\ell(w)\neq \sum_{w \sim u}\ell(w) $.
A graph may lack any $(0,1)$-additive labeling. We show that it is $
\mathbf{NP} $-complete to decide whether a $(0,1)$-additive labeling exists for
some families of graphs such as planar triangle-free graphs and perfect graphs.
For a graph $G$ with some $(0,1)$-additive labelings, the $(0,1)$-additive
number of $G$ is defined as $ \eta_{1} (G) = \min_{\ell \in \Gamma}\sum_{v\in
V(G)}\ell(v) $ where $\Gamma$ is the set of $(0,1)$-additive labelings of $G$.
We prove that given a planar graph that contains a $(0,1)$-additive labeling,
for all $ \varepsilon >0 $, approximating the $(0,1)$-additive number within $
n^{1-\varepsilon} $ is $ \mathbf{NP} $-hard.

May 2013

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25 Reads

Rudolph conjectures that for permutations $p$ and $q$ of the same length,
$A_n(p) \le A_n(q)$ for all $n$ if and only if the spine structure of $T(p)$ is
less than or equal to the spine structure of $T(q)$ in refinement order. We
prove one direction of this conjecture, by showing that if the spine structure
of $T(p)$ is less than or equal to the spine structure of $T(q)$, then $A_n(p)
\le A_n(q)$ for all $n$. We disprove the opposite direction by giving a
counterexample, and hence disprove the conjecture.

April 2009

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49 Reads

International audience
We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence.
Nous montrons que si la cardinalité d'un sous-ensemble de l'espace vectoriel à $(2k-1)$ dimensions sur un corps fini à $q$ éléments est $\gg q^{2k-1-\frac{1}{ 2k}}$, alors il contient une proportion non-nulle de tous les $k$-simplexes de congruence.

June 2013

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64 Reads

We show that permutations avoiding both of the (classical) patterns 4321 and
3241 have the algebraic generating function conjectured by Vladimir Kruchinin.

February 2010

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32 Reads

International audience
In this paper, we significantly improve a previous result by the same author showing the existence of a weakly universal cellular automaton with five states living in the hyperbolic $3D$-space. Here, we get such a cellular automaton with three states only.

February 2014

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56 Reads

Let $\mathbb{Z}_n$ denote the ring of integers modulo $n$. In this paper we
consider two extremal problems on permutations of $\mathbb{Z}_n$, namely, the
maximum size of a collection of permutations such that the sum of any two
distinct permutations in the collection is again a permutation, and the maximum
size of a collection of permutations such that the sum of any two distinct
permutations in the collection is not a permutation. Let the sizes be denoted
by $s(n)$ and $t(n)$ respectively. The case when $n$ is even is trivial in both
the cases, with $s(n)=1$ and $t(n)=n!$. For $n$ odd, we prove $s(n)\geq
(n\phi(n))/2^k$ where $k$ is the number of distinct prime divisors of $n$. When
$n$ is an odd prime we prove $s(n)\leq \frac{e^2}{\pi} n
((n-1)/e)^\frac{n-1}{2}$. For the second problem, we prove
$2^{(n-1)/2}.(\frac{n-1}{2})!\leq t(n)\leq 2^k.(n-1)!/\phi(n)$ when $n$ is odd.

July 2013

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255 Reads

We study natural bases for two constructions of the irreducible
representation of the symmetric group corresponding to $[n,n,n]$: the {\em
reduced web} basis associated to Kuperberg's combinatorial description of the
spider category; and the {\em left cell basis} for the left cell construction
of Kazhdan and Lusztig. In the case of $[n,n]$, the spider category is the
Temperley-Lieb category; reduced webs correspond to planar matchings, which are
equivalent to left cell bases. This paper compares the images of these bases
under classical maps: the {\em Robinson-Schensted algorithm} between
permutations and Young tableaux and {\em Khovanov-Kuperberg's bijection}
between Young tableaux and reduced webs.
One main result uses Vogan's generalized $\tau$-invariant to uncover a close
structural relationship between the web basis and the left cell basis.
Intuitively, generalized $\tau$-invariants refine the data of the inversion set
of a permutation. We define generalized $\tau$-invariants intrinsically for
Kazhdan-Lusztig left cell basis elements and for webs. We then show that the
generalized $\tau$-invariant is preserved by these classical maps. Thus, our
result allows one to interpret Khovanov-Kuperberg's bijection as an analogue of
the Robinson-Schensted correspondence.
Despite all of this, our second main result proves that the reduced web and
left cell bases are inequivalent; that is, these bijections are not
$S_{3n}$-equivariant maps.

October 2013

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17 Reads

We study Newton polytopes of cluster variables in type A_n cluster algebras,
whose cluster and coefficient variables are indexed by the diagonals and
boundary segments of a polygon. Our main results include an explicit
description of the affine hull and facets of the Newton polytope of the Laurent
expansion of any cluster variable, with respect to any cluster. In particular,
we show that every Laurent monomial in a Laurent expansion of a type A cluster
variable corresponds to a vertex of the Newton polytope. We also describe the
face lattice of each Newton polytope via an isomorphism with the lattice of
elementary subgraphs of the associated snake graph.

November 2010

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229 Reads

International audience
It is a well-known fact that the spacetime diagrams of some cellular automata have a fractal structure: for instance Pascal's triangle modulo $2$ generates a Sierpinski triangle. Explaining the fractal structure of the spacetime diagrams of cellular automata is a much explored topic, but virtually all of the results revolve around a special class of automata, whose main features include irreversibility, an alphabet with a ring structure and a rule respecting this structure, and a property known as being (weakly) $p$-Fermat. The class of automata that we study in this article fulfills none of these properties. Their cell structure is weaker and they are far from being $p$-Fermat, even weakly. However, they do produce fractal spacetime diagrams, and we will explain why and how. These automata emerge naturally from the field of quantum cellular automata, as they include the classical equivalent of the Clifford quantum cellular automata, which have been studied by the quantum community for several reasons. They are a basic building block of a universal model of quantum computation, and they can be used to generate highly entangled states, which are a primary resource for measurement-based models of quantum computing.

January 2009

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95 Reads

International audience
We show that the Kronecker coefficients indexed by two two―row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. These new formulas are obtained from analogous formulas for the corresponding reduced Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced Kronecker coefficients. As an application, we characterize all the Kronecker coefficients indexed by two two-row shapes that are equal to zero. This allowed us to disprove a conjecture of Mulmuley about the behavior of the stretching functions attached to the Kronecker coefficients.
Nous démontrons que les coefficients de Kronecker indexés par deux partitions de longueur au plus 2 sont donnés par des formules quasipolynomiales quadratiques dont les domaines de validité sont les cellules maximales d'un éventail. Des calculs simples nous donnent une description explicite des formules quasipolynomiales et de l'éventail associé. Ces nouvelles formulas sont obtenues de formules analogues pour les coefficients de Kronecker réduits correspondants et au moyen d'une formule reconstruisant les coefficients de Kronecker à partir des coefficients de Kronecker réduits. Une application est la caractérisation exacte de tous les coefficients de Kronecker non―nuls indexés par deux partitions de longueur au plus deux. Ceci nous a permis de réfuter une conjecture de Mulmuley au sujet des fonctions de dilatations associées aux coefficients de Kronecker.

January 2012

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12 Reads

Using a recursive approach, we obtain a simple exact expression for the
L^2-distance from the limit in R\'egnier's (1989) classical limit theorem for
the number of key comparisons required by QuickSort. A previous study by Fill
and Janson (2002) using a similar approach found that the d_2-distance is of
order between n^{-1} log n and n^{-1/2}, and another by Neininger and
Ruschendorf (2002) found that the Zolotarev zeta_3-distance is of exact order
n^{-1} log n. Our expression reveals that the L^2-distance is asymptotically
equivalent to (2 n^{-1} ln n)^{1/2}.

November 2014

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24 Reads

Inspired by M. Haiman's Operator Theorem, we study $\frak{S}_n$-modules of
polynomials in $\ell$ sets of $n$ variables, generated by a given homogeneous
diagonally symmetric polynomial $f$. These modules are closed by taking partial
derivatives, and generalized $\frak{S}_n$-invariants polarization operators. We
completely classify these modules (according to Frobenius transform) when they
are generated by degree 2 and degree 3 homogeneous symmetric polynomials. For
the classification of modules associated to homogeneous degree 3 symmetric
polynomials we introduce the notion of $n$-exception and we give an interesting
conjecture to characterise this notion. We compute general formulas for the
vector-graded Frobenius transform of $\frak{S}_n$-modules generated by degree 4
and degree 5 polynomials that seems to be universal.

January 2012

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16 Reads

International audience
In a continuous-time setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by $\texttt{QuickSort}$, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable $Y$—not even that it is nondegenerate. We establish the nondegeneracy of $Y$. The proof is perhaps surprisingly difficult.

July 2011

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27 Reads

In a biased weak $(a,b)$ polyform achievement game, the maker and the breaker
alternately mark $a,b$ previously unmarked cells on an infinite board,
respectively. The maker's goal is to mark a set of cells congruent to a
polyform. The breaker tries to prevent the maker from achieving this goal. A
winning maker strategy for the $(a,b)$ game can be built from winning
strategies for games involving fewer marks for the maker and the breaker. A new
type of breaker strategy called the priority strategy is introduced. The
winners are determined for all $(a,b)$ pairs for polyiamonds and polyominoes up
to size four.

November 2014

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9 Reads

International audience
We give an exact enumerative formula for the minimal acyclic deterministic finite automata. This formula is obtained from a bijection between a family of generalized parking functions and the transitions functions of acyclic automata.
On donne une formule d’énumération exacte des automates finites déterministes acycliques minimaux. Cetteformule s’obtient à partir d’une bijection entre une famille fonctions de parking généralisées et les fonctions detransitions des automates acycliques.

March 2005

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13 Reads

International audience
Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on $m$-ary search trees on $n$ keys with toll sequence $(i) n^α$ with $α ≥ 0 (α =0$ and $α =1$ correspond roughly to the space requirement and total path length, respectively); $(ii) ln \binom{n} {m-1}$, which corresponds to the so-called shape functional; and $(iii) $$1$$_{n=m-1}$, which corresponds to the number of leaves.

May 2014

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27 Reads

For planar graphs, we consider the problems of \emph{list edge coloring} and
\emph{list total coloring}. Edge coloring is the problem of coloring the edges
while ensuring that two edges that are adjacent receive different colors. Total
coloring is the problem of coloring the edges and the vertices while ensuring
that two edges that are adjacent, two vertices that are adjacent, or a vertex
and an edge that are incident receive different colors. In their list
extensions, instead of having the same set of colors for the whole graph, every
vertex or edge is assigned some set of colors and has to be colored from it. A
graph is minimally edge or total choosable if it is list edge
$\Delta$-colorable or list total $(\Delta+1)$-colorable, respectively, where
$\Delta$ is the maximum degree in the graph.
It is already known that planar graphs with $\Delta\geq 8$ and no triangle
adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and
that planar graphs with $\Delta\geq 7$ and no triangle sharing a vertex with a
$C_4$ or no triangle adjacent to a $C_k$ ($\forall 3 \leq k \leq 6$) are
minimally total colorable (Wang Wu 2011). We strengthen here these results and
prove that planar graphs with $\Delta\geq 7$ and no triangle adjacent to a
$C_4$ are minimally edge and total choosable.

September 2009

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117 Reads

We give a new expression for the expected number of inversions in the product of n random adjacent transpositions in the symmetric group S_{m+1}. We then derive from this expression the asymptotic behaviour of this number when n scales with m in various ways. Our starting point is an equivalence, due to Eriksson et al., with a problem of weighted walks confined to a triangular area of the plane.

May 2008

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88 Reads

Graphs and Algorithms
International audience
A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions.

March 2014

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78 Reads

By the algorithm implemented in the paper [2] by Akiyama-Lee, we have
examined pure discrete spectrum for some special cases of self-affine tilings.

September 2010

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9 Reads

International audience
We show the $q$-analog of a well-known result of Farahat and Higman: in the center of the Iwahori-Hecke algebra $\mathscr{H}_{n,q}$, if $(a_{\lambda \mu}^ν (n,q))_ν$ is the set of structure constants involved in the product of two Geck-Rouquier conjugacy classes $\Gamma_{\lambda, n}$ and $\Gamma_{\mu,n}$, then each coefficient $a_{\lambda \mu}^ν (n,q)$ depend on $n$ and $q$ in a polynomial way. Our proof relies on the construction of a projective limit of the Hecke algebras; this projective limit is inspired by the Ivanov-Kerov algebra of partial permutations.
Nous démontrons le $q$-analogue d'un résultat bien connu de Farahat et Higman : dans le centre de l'algèbre d'Iwahori-Hecke $\mathscr{H}_{n,q}$, si $(a_{\lambda \mu}^ν (n,q))_ν$ est l'ensemble des constantes de structure mises en jeu dans le produit de deux classes de conjugaison de Geck-Rouquier $\Gamma_{\lambda, n}$ et $\Gamma_{\mu,n}$, alors chaque coefficient $a_{\lambda \mu}^ν (n,q)$ dépend de façon polynomiale de $n$ et de $q$. Notre preuve repose sur la construction d'une limite projective des algèbres d'Hecke ; cette limite projective est inspirée de l'algèbre d'Ivanov-Kerov des permutations partielles.

November 2010

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19 Reads

International audience
This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $ν$, gives the spectral distribution of some random matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when $ν =(n)$ in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some permuted forests.
Cet article est dédié à l'évaluation des séries génératrices des coefficients de connexion des classes doubles (cosets) du groupe hyperoctaédral. Hanlon, Stanley, Stembridge (1992) ont montré que ces séries indexées par une partition $ν$ donnent la distribution spectrale de certaines matrices aléatoires jouant un rôle important dans la théorie des matrices aléatoires. Nous fournissons une évaluation explicite de ces séries dans le cas $ν =(n)$ en termes de monômes symétriques. Notre développement est fondé sur une interprétation des coefficients de connexion en termes d'hypercartes localement orientables et sur une nouvelle bijection entre les hypercartes localement orientables partitionnées et certaines forêts permutées.

August 2010

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64 Reads

International audience
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co-multiplication laws, thereby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the concepts of composition and decomposition with the emphasis on combinatorial origin of the ensuing algebraic constructions.

July 2013

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40 Reads

Combinatorics
International audience
A non-commutative, planar, Hopf algebra of planar rooted trees was defined independently by one of the authors in Foissy (2002) and by R. Holtkamp in Holtkamp (2003). In this paper we propose such a non-commutative Hopf algebra for graphs. In order to define a non-commutative product we use a quantum field theoretical (QFT) idea, namely the one of introducing discrete scales on each edge of the graph (which, within the QFT framework, corresponds to energy scales of the associated propagators). Finally, we analyze the associated quadri-coalgebra and codendrifrom structures.

August 2008

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37 Reads

M.-P. Sch\"utzenberger asked to determine the support of the free Lie algebra ${\mathcal L}_{{\mathbb Z}_{m}}(A)$ on a finite alphabet $A$ over the ring ${\mathbb Z}_{m}$ of integers $\bmod m$ and all the corresponding pairs of twin and anti-twin words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We study these problems using the adjoint endomorphism $l^{*}$ of the left normed Lie bracketing $l$ of ${\mathcal L}_{{\mathbb Z}_{m}}(A)$. Calculating $l^{*}(w)$ via all factors of a given word $w$ of fixed length and the shuffle product, we recover the result of Duchamp and Thibon $(1989)$ for the support of the free Lie ring in a much more natural way. We rephrase these problems, for words of length $n$, in terms of the action of the left normed multi-linear Lie bracketing $l_{n}$ of ${\mathcal L}_{{\mathbb Z}_{m}}(A)$ - viewed as an element of the group ring of the symmetric group ${\mathcal S}_{n}$ - on $\lambda$-tabloids, where $\lambda$ is a partition of $n$. For words $w$ in two letters, represented by a subset $I$ of $[n] = \{1, 2, ..., n \}$, this leads us to the {\em Pascal descent polynomial} $p_{n}(I)$, a particular commutative multi-linear polynomial which equals to a signed binomial coefficient when $|I| = 1$ and allows us to obtain a sufficient condition on $n$ and $I$ in order that $w$ lies in ${\mathcal L}_{{\mathbb Z}_{m}}(A)$. We also have a particular conjecture for twin and anti-twin words for the free Lie ring and show that it is enough to be checked for $|A| = 2$.