bluuebunny/crossref_metadata_2025 huggingface dataset:
p-box: a new graph model
Authors: Soto Mauricio , Thraves-Caro Christopher
Published: August 2021

Graph Theory

In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.
Some undecidable problems about the trace-subshift associated to a Turing machine
Authors: Gajardo Anahí , Ollinger Nicolas , Torres-Avilés Rodrigo
Published: August 2021

We consider three problems related to dynamics of one-tape Turing machines: Existence of blocking configurations, surjectivity in the trace, and entropy positiveness. In order to address them, a reversible two-counter machine is simulated by a reversible Turing machine on the right side of its tape. By completing the machine in different ways, we prove that none of the former problems is decidable. In particular, the problems about blocking configurations and entropy are shown to be undecidable for the class of reversible Turing machines.
Partitions of an Integer into Powers
Authors: Latapy Matthieu
Published: August 2021

In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.
On Minimal Strings Containing the Elements of S_n by Decimation
Authors: Erra Robert , Lygeros Nik , Stewart Nigel
Published: August 2021

The permutations by decimation problem is thought to be applicable to computer graphics, and raises interesting theoretical questions in combinatory theory.We present the results of some theoretical and practical investigation into this problem.We show that sequences of this form are $O(n^2)$ in length, but finding optimal solutions can be difficult.
Dynamics of the Picking transformation on integer partitions
Authors: Phan Thi Ha Duong , Thierry Eric
Published: August 2021

This paper studies a conservative transformation defined on families of finite sets. It consists in removing one element from each set and adding a new set composed of the removed elements. This transformation is conservative in the sense that the union of all sets of the family always remains the same.We study the dynamical process obtained when iterating this deterministic transformation on a family of sets and we focus on the evolution of the cardinalities of the sets of the family. This point of view allows to consider the transformation as an application defined on the set of all partitions of a fixed integer (which is the total number of elements in the sets).We show that iterating this particular transformation always leads to a heterogeneous distribution of the cardinalities, where almost all integers within an interval are represented.We also tackle some issues concerning the structure of the transition graph which sums up the whole dynamics of this process for all partitions of a fixed integer.
On P_4-tidy graphs
Authors: Giakoumakis V. , Roussel F. , Thuillier H.
Published: August 2021

We study the P_4-tidy graphs, a new class defined by Rusu [30] in order to illustrate the notion of P_4-domination in perfect graphs. This class strictly contains the P_4-extendible graphs and the P_4-lite graphs defined by Jamison & Olariu in [19] and [23] and we show that the P_4-tidy graphs and P_4-lite graphs are closely related. Note that the class of P_4-lite graphs is a class of brittle graphs strictly containing the P_4-sparse graphs defined by Hoang in [14]. McConnel & Spinrad [2] and independently Cournier & Habib [5] have shown that the modular decomposition tree of any graph is computable in linear time. For recognizing in linear time P_4-tidy graphs, we apply a method introduced by Giakoumakis in [9] and Giakoumakis & Fouquet in [6] using modular decomposition of graphs and we propose linear algorithms for optimization problems on such graphs, as clique number, stability number, chromatic number and scattering number. We show that the Hamiltonian Path Problem is linear for this class of graphs. Our study unifies and generalizes previous results of Jamison & Olariu ([18], [21], [22]), Hochstattler & Schindler[16], Jung [25] and Hochstattler & Tinhofer [15].
Lyndon factorization of the Thue-Morse word and its relatives
Authors: Ido Augustin , Melançon Guy
Published: August 2021

We compute the Lyndon factorization of the Thue-Morse word. We also compute the Lyndon factorization of two related sequences involving morphisms that give rise to new presentations of these sequences.
A $t$-generalization for Schubert Representatives of the Affine Grassmannian
Authors: Dalal Avinash J. , Morse Jennifer
Published: August 2021

We introduce two families of symmetric functions with an extra parameter $t$ that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when $t=1$. The families are defined by a statistic on combinatorial objects associated to the type-$A$ affine Weyl group and their transition matrix with Hall-Littlewood polynomials is $t$-positive. We conjecture that one family is the set of $k$-atoms.

Nous présentons deux familles de fonctions symétriques dépendant d'un paramètre $t$ et dont les spécialisations à $t=1$ correspondent aux classes de Schubert dans la cohomologie et l'homologie des variétés Grassmanniennes affines. Les familles sont définies par des statistiques sur certains objets combinatoires associés au groupe de Weyl affine de type $A$ et leurs matrices de transition dans la base des polynômes de Hall-Littlewood sont $t$-positives. Nous conjecturons qu'une de ces familles correspond aux $k$-atomes.
Lattice of combinatorial Hopf algebras: binary trees with multiplicities
Authors: Priez Jean-Baptiste
Published: August 2021

In a first part, we formalize the construction of combinatorial Hopf algebras from plactic-like monoids using polynomial realizations. Thank to this construction we reveal a lattice structure on those combinatorial Hopf algebras. As an application, we construct a new combinatorial Hopf algebra on binary trees with multiplicities and use it to prove a hook length formula for those trees.

Dans une première partie, nous formalisons la construction d’algèbres de Hopf combinatoires à partir d’une réalisation polynomiale et de monoïdes de type monoïde plaxique. Grâce à cette construction, nous mettons à jour une structure de treillis sur ces algèbres de Hopf combinatoires. Comme application, nous construisons une nouvelle algèbre de Hopf sur des arbres binaires à multiplicités et on l’utilise pour démontrer une formule des équerressur ces arbres.
$0$-Hecke algebra action on the Stanley-Reisner ring of the Boolean algebra
Authors: Huang Jia
Published: August 2021

We define an action of the $0$-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their $(q,t)$-analogues introduced by Bergeron and Zabrocki. We also obtain multivariate quasisymmetric function identities, which specialize to a result of Garsia and Gessel on the generating function of the joint distribution of five permutation statistics.

Nous définissons une action de l’algèbre de Hecke-$0$ de type A sur l’anneau Stanley-Reisner de l’algèbre de Boole. En étudiant cette action, on obtient une famille de fonctions symétriques non commutatives multivariées, qui se spécialisent pour les non commutatives fonctions de Hall-Littlewood symétriques et leur $(q,t)$-analogues introduits par Bergeron et Zabrocki. Nous obtenons également des identités de fonction quasisymmetrique multivariées, qui se spécialisent à la suite de Garsia et Gessel sur la fonction génératrice de la distribution conjointe de cinq statistiques de permutation.
Hopf Algebra of Sashes
Authors: Law Shirley
Published: August 2021

A general lattice theoretic construction of Reading constructs Hopf subalgebras of the Malvenuto-Reutenauer Hopf algebra (MR) of permutations. The products and coproducts of these Hopf subalgebras are defined extrinsically in terms of the embedding in MR. The goal of this paper is to find an intrinsic combinatorial description of a particular one of these Hopf subalgebras. This Hopf algebra has a natural basis given by permutations that we call Pell permutations. The Pell permutations are in bijection with combinatorial objects that we call sashes, that is, tilings of a 1 by n rectangle with three types of tiles: black 1 by 1 squares, white 1 by 1 squares, and white 1 by 2 rectangles. The bijection induces a Hopf algebra structure on sashes. We describe the product and coproduct in terms of sashes, and the natural partial order on sashes. We also describe the dual coproduct and dual product of the dual Hopf algebra of sashes.

Une construction générale dans la théorie des treillis dû à Reading construit des sous-algèbres de Hopf de l’algèbre de Hopf de permutations de Malvenuto et Reutenauer (MR). Les produits et coproduits de ces sous-algèbres de Hopf sont définis extrinsèquement en termes du plongement dans MR. Le but de cette communication est de trouver une description combinatoire intrinsèque d’une de ces sous-algèbres de Hopf en particulier. Cette algèbre Hopf a une base naturelle donnée par des permutations que nous appelons permutations Pell. Les permutations Pell sont en bijection avec des objets combinatoires que nous appelons écharpes, c’est-à-dire des pavages d’un rectangle 1-par-n avec trois espèces de tuiles : des carrés noirs 1-par-1, des carrés blancs 1-par-1, et des rectangles blancs 1-par-2. La bijection induit une structure d’algèbre de Hopf sur les écharpes. On décrit le produit et le coproduit en termes d’écharpes, et l’ordre partiel naturel sur les écharpes. On décrit également le coproduit dual et le produit dualde l’algèbre de Hopf dual des écharpes.
Walks in the Quarter Plane with Multiple Steps
Authors: Kauers Manuel , Yatchak Rika
Published: August 2021

We extend the classification of nearest neighbour walks in the quarter plane to models in which multiplicities are attached to each direction in the step set. Our study leads to a small number of infinite families that completely characterize all the models whose associated group is D4, D6, or D8. These families cover all the models with multiplicites 0, 1, 2, or 3, which were experimentally found to be D-finite — with three noteworthy exceptions.

Nous étendons la classification des marches aux plus proches voisins dans le quart de plan à des modèles dans lesquels une multiplicité est attachée à chaque direction de l’ensemble des pas. Notre étude identifie un petit nombre de familles infinies qui caractérisent complétement tous les modèles dont le groupe est D4, D6 ou D8. Ces familles contiennent tous les modèles à multiplicités 0, 1, 2 ou 3 dont il a été prouvé expérimentalement qu’ils étaientD-finis —avec trois exceptions notables.
The number of directed $k$-convex polyominoes
Authors: Boussicault Adrien , Rinaldi Simone , Socci Samanta
Published: August 2021

We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed $k$-convex polyominoes.We show it is a rational function and we study its asymptotic behavior.

Nous présentons une nouvelle méthode générique pour obtenir facilement et rapidement les fonctions génératrices des polyominos dirigés convexes avec différentes combinaisons de statistiques : hauteur, largeur, longueur de la dernière ligne/colonne et nombre de coins. La méthode peut être utilisée pour énumérer différentes familles de polyominos dirigés convexes: les polyominos symétriques, les polyominos parallélogrammes. De cette façon, nouscalculons la fonction génératrice des polyominos dirigés $k$-convexes, nous montrons qu’elle est rationnelle et nous étudions son comportement asymptotique.
Enumeration of minimal acyclic automata via generalized parking functions
Authors: Priez Jean-Baptiste
Published: August 2021

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.
Generalised cluster algebras and $q$-characters at roots of unity
Authors: Gleitz Anne-Sophie
Published: August 2021

Shapiro and Chekhov (2011) have introduced the notion of generalised cluster algebra; we focus on an example in type $C_n$. On the other hand, Chari and Pressley (1997), as well as Frenkel and Mukhin (2002), have studied the restricted integral form $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ of a quantum affine algebra $U_q(\widehat{\mathfrak{g}})$ where $q=ε$ is a root of unity. Our main result states that the Grothendieck ring of a tensor subcategory $C_{ε^\mathbb{z}}$ of representations of $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ is a generalised cluster algebra of type $C_{l−1}$, where $l$ is the order of $ε^2$. We also state a conjecture for $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$, and sketch a proof for $l=2$.

Shapiro et Chekhov (2011) ont introduit la notion d'algèbre amassée généralisée; nous étudions un exemple en type $C_n$. Par ailleurs, Chari et Pressley (1997), ainsi que Frenkel et Mukhin (2002), ont étudié la forme entière restreinte $U^{\mathtt{res}}_ε (\widehat{\mathfrak{g}})$ d'une algèbre affine quantique $U_q(\widehat{\mathfrak{g}})$ où $q=ε$ est une racine de l'unité. Notre résultat principal affirme que l'anneau de Grothendieck d'une sous-catégorie tensorielle $C_{ε^\mathbb{z}}$ de représentations de $U^{\mathtt{res}}_ε (L\mathfrak{sl}_2)$ est une algèbre amassée généralisée de type $C_{l−1}$, où $l$ est l'ordre de $ε^2$. Nous conjecturons une propriété similaire pour $U^{\mathtt{res}}_ε (L\mathfrak{sl}_3)$ et donnons un aperçu de la preuve pour $l=2$.
On bijections between monotone rooted trees and the comb basis
Authors: Liu Fu
Published: August 2021

Let $A$ be an $n$-element set. Let $\mathscr{L} ie_2(A)$ be the multilinear part of the free Lie algebra on $A$ with a pair of compatible Lie brackets, and $\mathscr{L} ie_2(A, i)$ the subspace of $\mathscr{L} ie_2(A)$ generated by all the monomials in $\mathscr{L} ie_2(A)$ with $i$ brackets of one type. The author and Dotsenko-Khoroshkin show that the dimension of $\mathscr{L} ie_2(A, i)$ is the size of $R_{A,i}$, the set of rooted trees on $A$ with $i$ decreasing edges. There are three families of bases known for $\mathscr{L} ie_2(A, i)$ the comb basis, the Lyndon basis, and the Liu-Lyndon basis. Recently, González D'León and Wachs, in their study of (co)homology of the poset of weighted partitions (which has close connection to $\mathscr{L} ie_2(A, i)$), asked whether there are nice bijections between $R_{A,i}$ and the comb basis or the Lyndon basis. We give a natural definition for " nice bijections " , and conjecture that there is a unique nice bijection between $R_{A,i}$ and the comb basis. We show the conjecture is true for the extreme cases where $i=0$, $n−1$.

Soit $A$ un ensemble à $n$ éléments. Soit $\mathscr{L} ie_2(A)$ la partie multilinéaire de l'algèbre de Lie libre sur $A$ avec une paire de crochets de Lie compatibles et $\mathscr{L} ie_2(A, i)$ le sous-espace de$\mathscr{L} ie_2(A)$ généré par tous les monômes en $\mathscr{L} ie_2(A)$ avec $i$ supports d'un même type. L'auteur et Dotsenko-Khoroshkin montrent que la dimension de $\mathscr{L} ie_2(A, i)$ est la taille de la $R_{A,i}$, l'ensemble des arbres enracinés sur $A$ avec $i$ arêtes décroissantes. Il y a trois familles de bases connues pour $\mathscr{L} ie_2(A, i)$ : la base de peigne, la base Lyndon, et la base Liu-Lyndon. Récemment, Gonzalez, D' Léon et Wachs, dans leur étude de (co)-homologie de la poset des partitions pondérés, ont demandé si il y a des bijections jolies entre$R_{A,i}$, et la base de peigne ou la base Lyndon. Nous donnons une définition naturelle de "bijection jolie " , et un conjecture qu'il y a une seule bijection jolie entre $R_{A,i}$, et la base de peigne. Nous montrons que la conjecture est vraie pour les cas extrêmes: $i = 0$, et $n − 1$.
A note on domino treewidth
Authors: Bodlaender Hans L.
Published: August 2021

In [DO95], Ding and Oporowski proved that for every k, and d, there exists a constant c_k,d, such that every graph with treewidth at most k and maximum degree at most d has domino treewidth at most c_k,d. This note gives a new simple proof of this fact, with a better bound for c_k,d, namely (9k+7)d(d+1) -1. It is also shown that a lower bound of Ω (kd) holds: there are graphs with domino treewidth at least 1/12 × kd-1, treewidth at most k, and maximum degree at most d, for many values k and d. The domino treewidth of a tree is at most its maximum degree.
A weakly universal cellular automaton in the hyperbolic $3D$ space with three states
Authors: Margenstern Maurice
Published: August 2021

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.
Block-sequential update schedules and Boolean automata circuits
Authors: Goles Eric , Noual Mathilde
Published: August 2021

Our work is set in the framework of complex dynamical systems and, more precisely, that of Boolean automata networks modeling regulation networks. We study how the choice of an update schedule impacts on the dynamics of such a network. To do this, we explain how studying the dynamics of any network updated with an arbitrary block-sequential update schedule can be reduced to the study of the dynamics of a different network updated in parallel. We give special attention to networks whose underlying structure is a circuit, that is, Boolean automata circuits. These particular and simple networks are known to serve as the "engines'' of the dynamics of arbitrary regulation networks containing them as sub-networks in that they are responsible for their variety of dynamical behaviours. We give both the number of attractors of period $p$, $\forall p\in \mathbb{N}$ and the total number of attractors in the dynamics of Boolean automata circuits updated with any block-sequential update schedule. We also detail the variety of dynamical behaviours that such networks may exhibit according to the update schedule.
An optimal cardinality estimation algorithm based on order statistics and its full analysis
Authors: Lumbroso Jérémie
Published: August 2021

Building on the ideas of Flajolet and Martin (1985), Alon et al. (1987), Bar-Yossef et al. (2002), Giroire (2005), we develop a new algorithm for cardinality estimation, based on order statistics which, according to Chassaing and Gerin (2006), is optimal among similar algorithms. This algorithm has a remarkably simple analysis that allows us to take its $\textit{fine-tuning}$ and the $\textit{characterization of its properties}$ further than has been done until now. We prove that, asymptotically, it is $\textit{strictly unbiased}$ (contrarily to Probabilistic Counting, Loglog, Hyperloglog), we verify that its relative precision is about $1/\sqrt{m-2}$ when $m$ words of storage are used, and we fully characterize the limit law of the estimates it provides, in terms of gamma distribution―-this is the first such algorithm for which the limit law has been established. We also develop a Poisson analysis for the pre-asymptotic regime. In this way, we are able to devise a complete algorithm, covering all cardinalities ranges from $0$ to very large.
Bounded discrete walks
Authors: Banderier C. , Nicodème P.
Published: August 2021

This article tackles the enumeration and asymptotics of directed lattice paths (that are isomorphic to unidimensional paths) of bounded height (walks below one wall, or between two walls, for $\textit{any}$ finite set of jumps). Thus, for any lattice paths, we give the generating functions of bridges ("discrete'' Brownian bridges) and reflected bridges ("discrete'' reflected Brownian bridges) of a given height. It is a new success of the "kernel method'' that the generating functions of such walks have some nice expressions as symmetric functions in terms of the roots of the kernel. These formulae also lead to fast algorithms for computing the $n$-th Taylor coefficients of the corresponding generating functions. For a large class of walks, we give the discrete distribution of the height of bridges, and show the convergence to a Rayleigh limit law. For the family of walks consisting of a $-1$ jump and many positive jumps, we give more precise bounds for the speed of convergence. We end our article with a heuristic application to bioinformatics that has a high speed-up relative to previous work.
Multi-dimensional Boltzmann Sampling of Languages
Authors: Bodini Olivier , Ponty Yann
Published: August 2021

We address the uniform random generation of words from a context-free language (over an alphabet of size $k$), while constraining every letter to a targeted frequency of occurrence. Our approach consists in a multidimensional extension of Boltzmann samplers. We show that, under mostly $\textit{strong-connectivity}$ hypotheses, our samplers return a word of size in $[(1- \epsilon)n, (1+ \epsilon)n]$ and exact frequency in $\mathcal{O}(n^{1+k/2})$ expected time. Moreover, if we accept tolerance intervals of width in $\Omega (\sqrt{n})$ for the number of occurrences of each letters, our samplers perform an approximate-size generation of words in expected $\mathcal{O}(n)$ time. We illustrate our approach on the generation of Tetris tessellations with uniform statistics in the different types of tetraminoes.
The cluster and dual canonical bases of Z [x_11, ..., x_33] are equal
Authors: Rhoades Brendon
Published: August 2021

The polynomial ring $\mathbb{Z}[x_{11}, . . . , x_{33}]$ has a basis called the dual canonical basis whose quantization facilitates the study of representations of the quantum group $U_q(\mathfrak{sl}3(\mathbb{C}))$. On the other hand, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ inherits a basis from the cluster monomial basis of a geometric model of the type $D_4$ cluster algebra. We prove that these two bases are equal. This extends work of Skandera and proves a conjecture of Fomin and Zelevinsky. This also provides an explicit factorization of the dual canonical basis elements of $\mathbb{Z}[x_{11}, . . . , x_{33}]$ into irreducible polynomials.

L'anneau de polynômes $\mathbb{Z}[x_{11}, . . . , x_{33}]$ a une base appelée base duale canonique, et dont une quantification facilite l'étude des représentations du groupe quantique $U_q(\mathfrak{sl}3(\mathbb{C}))$. D'autre part, $\mathbb{Z}[x_{11}, . . . , x_{33}]$ admet une base issue de la base des monômes d'amas de l'algèbre amassée géométrique de type $D_4$. Nous montrons que ces deux bases sont égales. Ceci prolonge les travaux de Skandera et démontre une conjecture de Fomin et Zelevinsky. Ceci fournit également une factorisation explicite en polynômes irréductibles des éléments de la base duale canonique de $\mathbb{Z}[x_{11}, . . . , x_{33}]$ .
The # product in combinatorial Hopf algebras
Authors: Aval Jean-Christophe , Novelli Jean-Christophe , Thibon Jean-Yves
Published: August 2021

We show that the # product of binary trees introduced by Aval and Viennot (2008) is in fact defined at the level of the free associative algebra, and can be extended to most of the classical combinatorial Hopf algebras.

Nous montrons que le produit # introduit par Aval et Viennot (2008) est défini au niveau de l'algèbre associative libre, et peut être étendu à la plupart des algèbres de Hopf combinatoires classiques.
Algebraic and combinatorial structures on Baxter permutations
Authors: Giraudo Samuele
Published: August 2021

We give a new construction of a Hopf subalgebra of the Hopf algebra of Free quasi-symmetric functions whose bases are indexed by objects belonging to the Baxter combinatorial family (\emphi.e. Baxter permutations, pairs of twin binary trees, \emphetc.). This construction relies on the definition of the Baxter monoid, analog of the plactic monoid and the sylvester monoid, and on a Robinson-Schensted-like insertion algorithm. The algebraic properties of this Hopf algebra are studied. This Hopf algebra appeared for the first time in the work of Reading [Lattice congruences, fans and Hopf algebras, \textitJournal of Combinatorial Theory Series A, 110:237–273, 2005].

Nous proposons une nouvelle construction d'une sous-algèbre de Hopf de l'algèbre de Hopf des fonctions quasi-symétriques libres dont les bases sont indexées par les objets de la famille combinatoire de Baxter (\emphi.e. permutations de Baxter, couples d'arbres binaires jumeaux, \emphetc.). Cette construction repose sur la définition du mono\"ıde de Baxter, analogue du mono\"ıde plaxique et du mono\"ıde sylvestre, et d'un algorithme d'insertion analogue à l'algorithme de Robinson-Schensted. Les propriétés algébriques de cette algèbre de Hopf sont étudiées. Cette algèbre de Hopf est apparue pour la première fois dans le travail de Reading [Lattice congruences, fans and Hopf algebras, \textitJournal of Combinatorial Theory Series A, 110:237–273, 2005].
The topological entropy of iterated piecewise affine maps is uncomputable
Authors: Koiran Pascal
Published: August 2021

We show that it is impossible to compute (or even to approximate) the topological entropy of a continuous piecewise affine function in dimension four. The same result holds for saturated linear functions in unbounded dimension. We ask whether the topological entropy of a piecewise affine function is always a computable real number, and conversely whether every non-negative computable real number can be obtained as the topological entropy of a piecewise affine function. It seems that these two questions are also open for cellular automata.
Matrices with restricted entries and q-analogues of permutations (extended abstract)
Authors: Lewis Joel Brewster , Liu Ricky Ini , Morales Alejandro H. , Panova Greta , Sam Steven V , Zhang Yan
Published: August 2021

We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a $q$-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank. Finally, we provide a brief exposition of polynomiality results for enumeration questions related to those mentioned, and give several open questions.

Nous étudions certaines fonctions qui comptent des matrices à coefficients dans un corps fini d'un rang donné ayant certaines entrées égales à zéro. Nous montrons que ces matrices sont des $q$-analogues des permutations avec certaines valeurs restreintes, et nous obtenons une formule simple et fermée pour calculer le nombre de matrices inversibles avec zéro sur toute la diagonale. De plus nous donnons des récursions pour énumérer par le rang les matrices et les matrices symétriques avec des zéros sur la diagonale. Pour finir, nous faisons un exposé concis des résultats sur la polynomialité des fonctions énumératives liées à celles qui sont mentionnées antérieurement, et nous incluons plusieurs questions ouvertes.
Projective subdynamics and universal shifts
Authors: Guillon Pierre
Published: August 2021

We study the projective subdynamics of two-dimensional shifts of finite type, which is the set of one-dimensional configurations that appear as columns in them. We prove that a large class of one-dimensional shifts can be obtained as such, namely the effective subshifts which contain positive-entropy sofic subshifts. The proof involves some simple notions of simulation that may be of interest for other constructions. As an example, it allows us to prove the undecidability of all non-trivial properties of projective subdynamics.
The weighted words collector
Authors: Du Boisberranger Jérémie , Gardy Danièle , Ponty Yann
Published: August 2021

We consider the word collector problem, i.e. the expected number of calls to a random weighted generator before all the words of a given length in a language are generated. The originality of this instance of the non-uniform coupon collector lies in the, potentially large, multiplicity of the words/coupons of a given probability/composition. We obtain a general theorem that gives an asymptotic equivalent for the expected waiting time of a general version of the Coupon Collector. This theorem is especially well-suited for classes of coupons featuring high multiplicities. Its application to a given language essentially necessitates knowledge on the number of words of a given composition/probability. We illustrate the application of our theorem, in a step-by-step fashion, on four exemplary languages, whose analyses reveal a large diversity of asymptotic waiting times, generally expressible as $\kappa \cdot m^p \cdot (\log{m})^q \cdot (\log \log{m})^r$, for $m$ the number of words, and $p, q, r$ some positive real numbers.
Simply generated trees, conditioned Galton―Watson trees, random allocations and condensation: Extended abstract
Authors: Janson Svante
Published: August 2021

We give a unified treatment of the limit, as the size tends to infinity, of random simply generated trees, including both the well-known result in the standard case of critical Galton-Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton-Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton-Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree. The random infinite limit tree can in all cases be constructed by a modified Galton-Watson process. In the standard case of a critical Galton-Watson tree, the limit tree has an infinite "spine", where the offspring distribution is size-biased. In the other cases, the spine has finite length and ends with a vertex with infinite degree. A node of infinite degree in the limit corresponds to the existence of one node with very high degree in the finite random trees; in physics terminology, this is a type of condensation. In simple cases, there is one node with a degree that is roughly a constant times the number of nodes, while all other degrees are much smaller; however, more complicated behaviour is also possible. The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding results for this model.
Synthesis of space-time optimal systolic algorithms for the Cholesky factorization
Authors: Tayou Djamegni Clémentin
Published: August 2021

In this paper we study the synthesis of space-time optimal systolic arrays for the Cholesky Factorization (CF). First, we discuss previous allocation methods and their application to CF. Second, stemming from a new allocation method we derive a space-time optimal array, with nearest neighbor connections, that requires 3N + Θ (1) time steps and N^2/8 + Θ (N) processors, where N is the size of the problem. The number of processors required by this new design improves the best previously known bound, N^2/6 + Θ (N), induced by previous allocation methods. This is the first contribution of the paper. The second contribution stemms from the fact that the paper also introduces a new allocation method that suggests to first perform clever index transformations on the initial dependence graph of a given system of uniform recurrent equations before applying the weakest allocation method, the projection method.
The down operator and expansions of near rectangular k-Schur functions
Authors: Berg Chris , Saliola Franco , Serrano Luis
Published: August 2021

We prove that the Lam-Shimozono ``down operator'' on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of ``near rectangles'' in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood–Richardson coefficients.

Nous montrons que l’opérateur ``down'', défini par Lam et Shimozono sur le groupe de Weyl affine, induit une dérivation de la sous-algèbre affine de Fomin-Stanley de l'algèbre affine de nilCoxeter. Nous employons cette dérivation pour vérifier une conjecture de Berg, Bergeron, Pon et Zabrocki sur l'expansion des k-fonctions de Schur indexées par les partitions qui sont ``presque rectangles''. Par conséquent, nous obtenons une interprétation combinatoire des k-coefficients de Littlewood–Richardson correspondants.
On a Subposet of the Tamari Lattice
Authors: Csar Sebastian A. , Sengupta Rik , Suksompong Warut
Published: August 2021

We discuss some properties of a subposet of the Tamari lattice introduced by Pallo (1986), which we call the comb poset. We show that three binary functions that are not well-behaved in the Tamari lattice are remarkably well-behaved within an interval of the comb poset: rotation distance, meets and joins, and the common parse words function for a pair of trees. We relate this poset to a partial order on the symmetric group studied by Edelman (1989).

Nous discutons d'un subposet du treillis de Tamari introduit par Pallo. Nous appellons ce poset le comb poset. Nous montrons que trois fonctions binaires qui ne se comptent pas bien dans le trellis de Tamari se comptent bien dans un intervalle du comb poset : distance dans le trellis de Tamari, le supremum et l'infimum et les parsewords communs. De plus, nous discutons un rapport entre ce poset et un ordre partiel dans le groupe symétrique étudié par Edelman.
Coxeter-like complexes
Authors: Babson Eric , Reiner Victor
Published: August 2021

Motivated by the Coxeter complex associated to a Coxeter system (W,S), we introduce a simplicial regular cell complex Δ (G,S) with a G-action associated to any pair (G,S) where G is a group and S is a finite set of generators for G which is minimal with respect to inclusion. We examine the topology of Δ (G,S), and in particular the representations of G on its homology groups. We look closely at the case of the symmetric group S_n minimally generated by (not necessarily adjacent) transpositions, and their type-selected subcomplexes. These include not only the Coxeter complexes of type A, but also the well-studied chessboard complexes.
Area of Brownian Motion with Generatingfunctionology
Authors: Nguyên Thê Michel
Published: August 2021

This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.
A Random Walk Approach for Light Scattering in Material
Authors: Simon Klaus , Trachsler Beat
Published: August 2021

Understanding reflection is one of the key competences in graphic arts industry. A very popular approach was given by Kubelka andMunk [1931] who derived a simple relationship between the scattering and absorption coefficients and the overall reflectance. This paper presents an alternative approach which describes the behavior of light in matter as a special kind of random walk.
Constructing a sequence of random walks strongly converging to Brownian motion
Authors: Marchal Philippe
Published: August 2021

We give an algorithm which constructs recursively a sequence of simple random walks on $\mathbb{Z}$ converging almost surely to a Brownian motion. One obtains by the same method conditional versions of the simple random walk converging to the excursion, the bridge, the meander or the normalized pseudobridge.
A phase transition in the random transposition random walk
Authors: Berestycki Nathanael , Durrett Rick
Published: August 2021

Our work is motivated by Bourque-Pevzner's simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk in continuous time on the group of permutations on $n$ elements starting from the identity. Let $D_t$ be the minimum number of transpositions needed to go back to the identity element from the location at time $t$. $D_t$ undergoes a phase transition: for $0 < c ≤ 1$, the distance $D_cn/2 ~ cn/2$, i.e., the distance increases linearly with time; for $c > 1$, $D_cn/2 ~ u(c)n$ where u is an explicit function satisfying $u(x) < x/2$. Moreover we describe the fluctuations of $D_{cn/2}$ about its mean at each of the three stages (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the Erdős-Rényi random graph model.
Cache efficient simple dynamic programming
Authors: Cherng Cary , Ladner Richard E.
Published: August 2021

New cache-oblivious and cache-aware algorithms for simple dynamic programming based on Valiant's context-free language recognition algorithm are designed, implemented, analyzed, and empirically evaluated with timing studies and cache simulations. The studies show that for large inputs the cache-oblivious and cache-aware dynamic programming algorithms are significantly faster than the standard dynamic programming algorithm.
Analytic combinatorics for a certain well-ordered class of iterated exponential terms
Authors: Weiermann Andreas
Published: August 2021

The aim of this paper is threefold: firstly, to explain a certain segment of ordinals in terms which are familiar to the analytic combinatorics community, secondly to state a great many of associated problems on resulting count functions and thirdly, to provide some weak asymptotic for the resulting count functions. We employ for simplicity Tauberian methods. The analytic combinatorics community is encouraged to provide (maybe in joint work) sharper results in future investigations.
Application of data compression methods to hypothesis testing for ergodic and stationary processes
Authors: Ryabko Boris , Astola Jaakko
Published: August 2021

We show that data compression methods (or universal codes) can be applied for hypotheses testing in a framework of classical mathematical statistics. Namely, we describe tests, which are based on data compression methods, for the three following problems: i) identity testing, ii) testing for independence and iii) testing of serial independence for time series. Applying our method of identity testing to pseudorandom number generators, we obtained experimental results which show that the suggested tests are quite efficient.
Distribution-sensitive set multi-partitioning
Authors: Elmasry Amr
Published: August 2021

Given a set $\mathcal{S}$ with real-valued members, associated with each member one of two possible types; a multi-partitioning of $\mathcal{S}$ is a sequence of the members of $\mathcal{S}$ such that if $x,y \in \mathcal{S}$ have different types and $x < y$, $x$ precedes $y$ in the multi-partitioning of $\mathcal{S}$. We give two distribution-sensitive algorithms for the set multi-partitioning problem and a matching lower bound in the algebraic decision-tree model. One of the two algorithms can be made stable and can be implemented in place. We also give an output-sensitive algorithm for the problem.
Density of universal classes of series-parallel graphs
Authors: Nešetřil Jaroslav , Nigussie Yared
Published: August 2021

A class of graphs $\mathcal{C}$ ordered by the homomorphism relation is universal if every countable partial order can be embedded in $\mathcal{C}$. It was shown in [ZH] that the class $\mathcal{C_k}$ of $k$-colorable graphs, for any fixed $k≥3$, induces a universal partial order. In [HN1], a surprisingly small subclass of $\mathcal{C_3}$ which is a proper subclass of $K_4$-minor-free graphs $(\mathcal{G/K_4)}$ is shown to be universal. In another direction, a density result was given in [PZ], that for each rational number $a/b ∈[2,8/3]∪ \{3\}$, there is a $K_4$-minor-free graph with circular chromatic number equal to $a/b$. In this note we show for each rational number $a/b$ within this interval the class $\mathcal{K_{a/b}}$ of $0K_4$-minor-free graphs with circular chromatic number $a/b$ is universal if and only if $a/b ≠2$, $5/2$ or $3$. This shows yet another surprising richness of the $K_4$-minor-free class that it contains universal classes as dense as the rational numbers.
Infinite families of accelerated series for some classical constants by the Markov-WZ Method
Authors: Mohammed Mohamud
Published: August 2021

In this article we show the Markov-WZ Method in action as it finds rapidly converging series representations for a given hypergeometric series. We demonstrate the method by finding new representations for log(2),ζ (2) and ζ (3).
Weak Positional Games on Hypergraphs of Rank Three
Authors: Kutz Martin
Published: August 2021

In a weak positional game, two players, Maker and Breaker, alternately claim vertices of a hypergraph until either Maker wins by getting a complete edge or all vertices are taken without this happening, a Breaker win. For the class of almost-disjoint hypergraphs of rank three (edges with up to three vertices only and edge-intersections on at most one vertex) we show how to find optimal strategies in polynomial time. Our result is based on a new type of decomposition theorem which might lead to a better understanding of weak positional games in general.
Local chromatic number and topology
Authors: Simonyi Gábor , Tardos Gábor
Published: August 2021

The local chromatic number of a graph, introduced by Erdős et al., is the minimum number of colors that must appear in the closed neighborhood of some vertex in any proper coloring of the graph. This talk would like to survey some of our recent results on this parameter. We give a lower bound for the local chromatic number in terms of the lower bound of the chromatic number provided by the topological method introduced by Lovász. We show that this bound is tight in many cases. In particular, we determine the local chromatic number of certain odd chromatic Schrijver graphs and generalized Mycielski graphs. We further elaborate on the case of $4$-chromatic graphs and, in particular, on surface quadrangulations.
Maximal sets of integers not containing $k+1$ pairwise coprimes and having divisors from a specified set of primes
Authors: Blinovsky Vladimir
Published: August 2021

We find the formula for the cardinality of maximal set of integers from $[1,\ldots,n]$ which does not contain $k+1$ pairwise coprimes and has divisors from a specified set of primes. This formula is defined by the set of multiples of the generating set, which does not depend on $n$.
Improving the Gilbert-Varshamov bound for $q$-ary codes
Authors: Vu Van H. , Wu Lei
Published: August 2021

Given positive integers $q$, $n$ and $d$, denote by $A_q(n,d)$ the maximum size of a $q$-ary code of length $n$ and minimum distance $d$. The famous Gilbert-Varshamov bound asserts that $A_q(n,d+1) \geq q^n / V_q(n,d)$, where $V_q(n,d)=\sum_{i=0}^d \binom{n}{i}(q-1)^i$ is the volume of a $q$-ary sphere of radius $d$. Extending a recent work of Jiang and Vardy on binary codes, we show that for any positive constant $\alpha$ less than $(q-1)/q$ there is a positive constant $c$ such that for $d \leq \alpha n, A_q(n,d+1) \geq c \frac{q^n}{ V_q(n,d)}n$. This confirms a conjecture by Jiang and Vardy.
On the maximum average degree and the incidence chromatic number of a graph
Authors: Dolama Mohammad Hosseini , Sopena Eric
Published: August 2021

We prove that the incidence chromatic number of every 3-degenerated graph G is at most Δ (G)+4. It is known that the incidence chromatic number of every graph G with maximum average degree mad(G)<3 is at most Δ (G)+3. We show that when Δ (G) ≥ 5, this bound may be decreased to Δ (G)+2. Moreover, we show that for every graph G with mad(G)<22/9 (resp. with mad(G)<16/7 and Δ (G)≥ 4), this bound may be decreased to Δ (G)+2 (resp. to Δ (G)+1).
Efficient estimation of the cardinality of large data sets
Authors: Chassaing Philippe , Gerin Lucas
Published: August 2021

Giroire has recently proposed an algorithm which returns the $\textit{approximate}$ number of distinct elements in a large sequence of words, under strong constraints coming from the analysis of large data bases. His estimation is based on statistical properties of uniform random variables in $[0,1]$. In this note we propose an optimal estimation, using Kullback information and estimation theory.
Spanning trees of finite Sierpiński graphs
Authors: Teufl Elmar , Wagner Stephan
Published: August 2021

We show that the number of spanning trees in the finite Sierpiński graph of level $n$ is given by $\sqrt[4]{\frac{3}{20}} (\frac{5}{3})^{-n/2} (\sqrt[4]{540})^{3^n}$. The proof proceeds in two steps: First, we show that the number of spanning trees and two further quantities satisfy a $3$-dimensional polynomial recursion using the self-similar structure. Secondly, it turns out, that the dynamical behavior of the recursion is given by a $2$-dimensional polynomial map, whose iterates can be computed explicitly.
Density of truth in modal logics
Authors: Kostrzycka Zofia
Published: August 2021

The aim of this paper is counting the probability that a random modal formula is a tautology. We examine $\{ \to,\Box \}$ fragment of two modal logics $\mathbf{S5}$ and $\mathbf{S4}$ over the language with one propositional variable. Any modal formula written in such a language may be interpreted as a unary binary tree. As it is known, there are finitely many different formulas written in one variable in the logic $\mathbf{S5}$ and this is the key to count the proportion of tautologies of $\mathbf{S5}$ among all formulas. Although the logic $\mathbf{S4}$ does not have this property, there exist its normal extensions having finitely many non-equivalent formulas.