bluuebunny/crossref_metadata_2025 huggingface dataset:
Conditioned Galton-Watson trees do not grow
Authors: Janson Svante
Published: August 2021

An example is given which shows that, in general, conditioned Galton-Watson trees cannot be obtained by adding vertices one by one, while this can be done in some important but special cases, as shown by Luczak and Winkler.
Explicit computation of the variance of the number of maxima in hypercubes
Authors: Costermans Christian , Minh Hoang Ngoc
Published: August 2021

We present a combinatorial approach of the variance for the number of maxima in hypercubes. This leads to an explicit expression, in terms of Multiple Zeta Values, of the dominant term in the asymptotic expansion of this variance.Moreover, we get an algorithm to compute this expansion, and show that all coefficients occuring belong to the $\mathbb{Q}$-algebra generated by Multiple Zeta Values, and by Euler's constant $\gamma$.
Average depth in a binary search tree with repeated keys
Authors: Archibald Margaret , Clément Julien
Published: August 2021

Random sequences from alphabet $\{1, \ldots,r\}$ are examined where repeated letters are allowed. Binary search trees are formed from these, and the average left-going depth of the first $1$ is found. Next, the right-going depth of the first $r$ is examined, and finally a merge (or 'shuffle') operator is used to obtain the average depth of an arbitrary node, which can be expressed in terms of the left-going and right-going depths. The variance of each of these parameters is also found.
Multivariate generalizations of the Foata-Schützenberger equidistribution
Authors: Hivert Florent , Novelli Jean-Christophe , Thibon Jean-Yves
Published: August 2021

A result of Foata and Schützenberger states that two statistics on permutations, the number of inversions and the inverse major index, have the same distribution on a descent class. We give a multivariate generalization of this property: the sorted vectors of the Lehmer code, of the inverse majcode, and of a new code (the inverse saillance code), have the same distribution on a descent class, and their common multivariate generating function is a flagged ribbon Schur function.
Walking Cautiously Into the Collatz Wilderness: Algorithmically, Number Theoretically, Randomly
Authors: Belaga Edward G. , Mignotte Maurice
Published: August 2021

Building on theoretical insights and rich experimental data of our preprints, we present here new theoretical and experimental results in three interrelated approaches to the Collatz problem and its generalizations: \emphalgorithmic decidability, random behavior, and Diophantine representation of related discrete dynamical systems, and their \emphcyclic and divergent properties.
Concentration Properties of Extremal Parameters in Random Discrete Structures
Authors: Drmota Michael
Published: August 2021

The purpose of this survey is to present recent results concerning concentration properties of extremal parameters of random discrete structures. A main emphasis is placed on the height and maximum degree of several kinds of random trees. We also provide exponential tail estimates for the height distribution of scale-free trees.
Degree distribution of random Apollonian network structures and Boltzmann sampling
Authors: Darrasse Alexis , Soria Michèle
Published: August 2021

Random Apollonian networks have been recently introduced for representing real graphs. In this paper we study a modified version: random Apollonian network structures (RANS), which preserve the interesting properties of real graphs and can be handled with powerful tools of random generation. We exhibit a bijection between RANS and ternary trees, that transforms the degree of nodes in a RANS into the size of particular subtrees. The distribution of degrees in RANS can thus be analysed within a bivariate Boltzmann model for the generation of random trees, and we show that it has a Catalan form which reduces to a power law with an exponential cutoff: $α ^k k^{-3/2}$, with $α = 8/9$. We also show analogous distributions for the degree in RANS of higher dimension, related to trees of higher arity.
Online Bandwidth packing with symmetric distribution
Authors: Lelarge Marc
Published: August 2021

We consider the following stochastic bin packing process: the items arrive continuously over time to a server and are packed into bins of unit size according to an online algorithm. The unpacked items form a queue. The items have random sizes with symmetric distribution. Our first contribution identifies some monotonicity properties of the queueing system that allow to derive bounds on the queue size for First Fit and Best Fit algorithms. As a direct application, we show how to compute the stability region under very general conditions on the input process. Our second contribution is a study of the queueing system under heavy load. We show how the monotonicity properties allow one to derive bounds for the speed at which the stationary queue length tends to infinity when the load approaches one. In the case of Best Fit, these bounds are tight. Our analysis shows connections between our dynamic model, average-case results on the classical bin packing problem and planar matching problems.
Random permutations and their discrepancy process
Authors: Chapuy Guillaume
Published: August 2021

Let $\sigma$ be a random permutation chosen uniformly over the symmetric group $\mathfrak{S}_n$. We study a new "process-valued" statistic of $\sigma$, which appears in the domain of computational biology to construct tests of similarity between ordered lists of genes. More precisely, we consider the following "partial sums": $Y^{(n)}_{p,q} = \mathrm{card} \{1 \leq i \leq p : \sigma_i \leq q \}$ for $0 \leq p,q \leq n$. We show that a suitable normalization of $Y^{(n)}$ converges weakly to a bivariate tied down brownian bridge on $[0,1]^2$, i.e. a continuous centered gaussian process $X^{\infty}_{s,t}$ of covariance: $\mathbb{E}[X^{\infty}_{s,t}X^{\infty}_{s',t'}] = (min(s,s')-ss')(min(t,t')-tt')$.
The Height of List-tries and TST
Authors: Broutin N. , Devroye L.
Published: August 2021

We characterize the asymptotics of heights of the trees of de la Briandais and the ternary search trees (TST) of Bentley and Sedgewick. Our proof is based on a new analysis of the structure of tries that distinguishes the bulk of the tree, called the $\textit{core}$, and the long trees hanging down the core, called the $\textit{spaghettis}$.
Combinatorial Dominance Guarantees for Heuristic Algorithms
Authors: Berend Daniel , Skiena Steven S. , Twitto Yochai
Published: August 2021

An $f(n)$ $\textit{dominance bound}$ on a heuristic for some problem is a guarantee that the heuristic always returns a solution not worse than at least $f(n)$ solutions. In this paper, we analyze several heuristics for $\textit{Vertex Cover}$, $\textit{Set Cover}$, and $\textit{Knapsack}$ for dominance bounds. In particular, we show that the well-known $\textit{maximal matching}$ heuristic of $\textit{Vertex Cover}$ provides an excellent dominance bound. We introduce new general analysis techniques which apply to a wide range of problems and heuristics for this measure. Certain general results relating approximation ratio and combinatorial dominance guarantees for optimization problems over subsets are established. We prove certain limitations on the combinatorial dominance guarantees of polynomial-time approximation schemes (PTAS), and give inapproximability results for the problems above.
Optimal Prefix and Suffix Queries on Texts
Authors: Crochemore Maxime , Iliopoulos Costas S. , Rahman M. Sohel
Published: August 2021

In this paper, we study a restricted version of the position restricted pattern matching problem introduced and studied by Mäkinen and Navarro [Position-Restricted Substring Searching, LATIN 2006]. In the problem handled in this paper, we are interested in those occurrences of the pattern that lies in a suffix or in a prefix of the given text. We achieve optimal query time for our problem against a data structure which is an extension of the classic suffix tree data structure. The time and space complexity of the data structure is dominated by that of the suffix tree. Notably, the (best) algorithm by Mäkinen and Navarro, if applied to our problem, gives sub-optimal query time and the corresponding data structure also requires more time and space.
On expected number of maximal points in polytopes
Authors: Baryshnikov Yu.
Published: August 2021

We answer an old question: what are possible growth rates of the expected number of vector-maximal points in a uniform sample from a polytope.
On Correlation Polynomials and Subword Complexity
Authors: Gheorghiciuc Irina , Ward Mark Daniel
Published: August 2021

We consider words with letters from a $q-ary$ alphabet $\mathcal{A}$. The kth subword complexity of a word $w ∈\mathcal{A}^*$ is the number of distinct subwords of length $k$ that appear as contiguous subwords of $w$. We analyze subword complexity from both combinatorial and probabilistic viewpoints. Our first main result is a precise analysis of the expected $kth$ subword complexity of a randomly-chosen word $w ∈\mathcal{A}^n$. Our other main result describes, for $w ∈\mathcal{A}^*$, the degree to which one understands the set of all subwords of $w$, provided that one knows only the set of all subwords of some particular length $k$. Our methods rely upon a precise characterization of overlaps between words of length $k$. We use three kinds of correlation polynomials of words of length $k$: unweighted correlation polynomials; correlation polynomials associated to a Bernoulli source; and generalized multivariate correlation polynomials. We survey previously-known results about such polynomials, and we also present some new results concerning correlation polynomials.
Constructions for Clumps Statistics.
Authors: Bassino Frédérique , Clément Julien , Fayolle Julien , Nicodème Pierre
Published: August 2021

We consider a component of the word statistics known as clump; starting from a finite set of words, clumps are maximal overlapping sets of these occurrences. This object has first been studied by Schbath with the aim of counting the number of occurrences of words in random texts. Later work with similar probabilistic approach used the Chen-Stein approximation for a compound Poisson distribution, where the number of clumps follows a law close to Poisson. Presently there is no combinatorial counterpart to this approach, and we fill the gap here. We also provide a construction for the yet unsolved problem of clumps of an arbitrary finite set of words. In contrast with the probabilistic approach which only provides asymptotic results, the combinatorial method provides exact results that are useful when considering short sequences.
On the number of zero increments of random walks with a barrier
Authors: Iksanov Alex , Negadajlov Pavlo
Published: August 2021

Continuing the line of research initiated in Iksanov and Möhle (2008) and Negadajlov (2008) we investigate the asymptotic (as $n \to \infty$) behaviour of $V_n$ the number of zero increments before the absorption in a random walk with the barrier $n$. In particular, when the step of the unrestricted random walk has a finite mean, we prove that the number of zero increments converges in distribution. We also establish a weak law of large numbers for $V_n$ under a regular variation assumption.
Random Records and Cuttings in Split Trees: Extended Abstract
Authors: Holmgren Cecilia
Published: August 2021

We study the number of records in random split trees on $n$ randomly labelled vertices. Equivalently the number of random cuttings required to eliminate an arbitrary random split tree can be studied. After normalization the distributions are shown to be asymptotically $1$-stable. This work is a generalization of our earlier results for the random binary search tree which is one specific case of split trees. Other important examples of split trees include $m$-ary search trees, quadtrees, median of $(2k+1)$-trees, simplex trees, tries and digital search trees.
Analysis of an algorithm catching elephants on the Internet
Authors: Chabchoub Yousra , Fricker Christine , Meunier Frédéric , Tibi Danielle
Published: August 2021

The paper deals with the problem of catching the elephants in the Internet traffic. The aim is to investigate an algorithm proposed by Azzana based on a multistage Bloom filter, with a refreshment mechanism (called $\textit{shift}$ in the present paper), able to treat on-line a huge amount of flows with high traffic variations. An analysis of a simplified model estimates the number of false positives. Limit theorems for the Markov chain that describes the algorithm for large filters are rigorously obtained. The asymptotic behavior of the stochastic model is here deterministic. The limit has a nice formulation in terms of a $M/G/1/C$ queue, which is analytically tractable and which allows to tune the algorithm optimally.
Convergence to the coalescent and its relation to the time back to the most recent common ancestor
Authors: Möhle Martin
Published: August 2021

For the class of haploid exchangeable population models with non-overlapping generations and population size $N$ it is shown that, as $N$ tends to infinity, convergence of the time-scaled ancestral process to Kingman's coalescent and convergence in distribution of the scaled times back to the most recent common ancestor (MRCA) to the corresponding times back to the MRCA of the Kingman coalescent are equivalent. Extensions of this equivalence are derived for exchangeable population models being in the domain of attraction of a coalescent process with multiple collisions. The proofs are based on the property that the total rates of a coalescent with multiple collisions already determine the distribution of the coalescent. It is finally shown that similar results cannot be obtained for the full class of exchangeable coalescents allowing for simultaneous multiple collisions of ancestral lineages, essentially because the total rates do not determine the distribution of a general exchangeable coalescent.
Are even maps on surfaces likely to be bipartite?
Authors: Chapuy Guillaume
Published: August 2021

It is well known that a planar map is bipartite if and only if all its faces have even degree (what we call an even map). In this paper, we show that rooted even maps of positive genus $g$ chosen uniformly at random are bipartite with probability tending to $4^{−g}$ when their size goes to infinity. Loosely speaking, we show that each of the $2g$ fundamental cycles of the surface of genus $g$ contributes a factor $\frac{1}{2}$ to this probability.We actually do more than that: we obtain the explicit asymptotic behaviour of the number of even maps and bipartite maps of given genus with any finite set of allowed face degrees. This uses a generalisation of the Bouttier-Di Francesco-Guitter bijection to the case of positive genus, a decomposition inspired by previous works of Marcus, Schaeffer and the author, and some involved manipulations of generating series counting paths. A special case of our results implies former conjectures of Gao.
A Markov Chain Algorithm for determining Crossing Times through nested Graphs
Authors: Freiberg Uta , Thäle Christoph
Published: August 2021

According to the by now established theory developed in order to define a Laplacian or ― equivalently ― a Brownian motion on a nested fractal, one has to solve certain renormalization problems. In this paper, we present a Markov chain algorithm solving the problem for certain classes of simple fractals $K$ provided that there exists a unique Brownian motion and hence, a unique Laplacian on $K$.
The Online Specialization Problem
Authors: Hong Edwin S.
Published: August 2021

We study the online specialization problem, where items arrive in an online fashion for processing by one of n different methods. Each method has two costs: a processing cost (paid once for each item processed), and a set-up cost (paid only once, on the method's first use). There are n possible types of items; an item's type determines the set of methods available to process it. Each method has a different degree of specialization. Highly specialized methods can process few item types while generic methods may process all item types. This is a generalization of ski-rental and closely related to the capital investment problem of Y. Azar, Y. Bartal, E. Feuerstein, A. Fiat, S. Leonardi, and A. Rosen. On capital investment. In Algorithmica, 25(1):22-36, 1999.. We primarily study the case where method i+1 is always more specialized than method i and the set-up cost for a more specialized method is always higher than that of a less specialized method. We describe an algorithm with competitive ratio O(log(n)), and also show an Ω (log(n)) lower bound on the competitive ratio for this problem; this shows our ratio is tight up to constant factors.
The Sorting Order on a Coxeter Group
Authors: Armstrong Drew
Published: August 2021

Let $(W,S)$ be an arbitrary Coxeter system. For each sequence $\omega =(\omega_1,\omega_2,\ldots) \in S^{\ast}$ in the generators we define a partial order― called the $\omega \mathsf{-sorting order}$ ―on the set of group elements $W_{\omega} \subseteq W$ that occur as finite subwords of $\omega$ . We show that the $\omega$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and strong Bruhat orders on the group. Moreover, the $\omega$-sorting order is a "maximal lattice'' in the sense that the addition of any collection of edges from the Bruhat order results in a nonlattice. Along the way we define a class of structures called $\mathsf{supersolvable}$ $\mathsf{antimatroids}$ and we show that these are equivalent to the class of supersolvable join-distributive lattices.
Quasisymmetric Schur functions
Authors: Haglund James , Mason Sarah , Luoto Kurt , van Willigenburg Steph
Published: August 2021

We introduce a new basis for the algebra of quasisymmetric functions that naturally partitions Schur functions, called quasisymmetric Schur functions. We describe their expansion in terms of fundamental quasisymmetric functions and determine when a quasisymmetric Schur function is equal to a fundamental quasisymmetric function. We conclude by describing a Pieri rule for quasisymmetric Schur functions that naturally generalizes the Pieri rule for Schur functions.

Nous étudions une nouvelle base des fonctions quasisymétriques, les fonctions de quasiSchur. Ces fonctions sont obtenues en spécialisant les fonctions de Macdonald dissymétrique. Nous décrivons les compositions que donne une simple fonction quasisymétriques. Nous décrivons aussi une règle par certaines fonctions de Schur.
A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers
Authors: Méndez Miguel , Rodríguez Adolfo
Published: August 2021

We describe a combinatorial model for the $q$-analogs of the generalized Stirling numbers in terms of bugs and colonies. Using both algebraic and combinatorial methods, we derive explicit formulas, recursions and generating functions for these $q$-analogs. We give a weight preserving bijective correspondence between our combinatorial model and rook placements on Ferrer boards. We outline a direct application of our theory to the theory of dual graded graphs developed by Fomin. Lastly we define a natural $p,q$-analog of these generalized Stirling numbers.
Double Schubert polynomials for the classical Lie groups
Authors: Ikeda Takeshi , Mihalcea Leonardo , Naruse Hiroshi
Published: August 2021

For each infinite series of the classical Lie groups of type $B$, $C$ or $D$, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the corresponding flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. When indexed by maximal Grassmannian elements of the Weyl group, these polynomials are equal to the factorial analogues of Schur $Q$- or $P$-functions defined earlier by Ivanov.

Pour chaque série infinie des groupe de Lie classiques de type $B$,$C$ ou $D$, nous présentons une famille de polynômes indexées par de éléments de groupe de Weyl correspondant de rang infini. Ces polynômes représentent des classes de Schubert dans la cohomologie équivariante des variétés de drapeaux. Ils ont une certain propriété de stabilité, et ils étendent naturellement des polynômes Schubert (simples) de Billey et Haiman, que représentent des classes de Schubert dans la cohomologie non-équivariante. Quand ils sont indexées par des éléments Grassmanniennes de groupes de Weyl, ces polynômes sont égaux à des analogues factorielles de fonctions $Q$ et $P$ de Schur, étudiées auparavant par Ivanov.
Optimal L(h,k)-Labeling of Regular Grids
Authors: Calamoneri Tiziana
Published: August 2021

The L(h, k)-labeling is an assignment of non negative integer labels to the nodes of a graph such that 'close' nodes have labels which differ by at least k, and 'very close' nodes have labels which differ by at least h. The span of an L(h,k)-labeling is the difference between the largest and the smallest assigned label. We study L(h, k)-labelings of cellular, squared and hexagonal grids, seeking those with minimum span for each value of k and h ≥ k. The L(h,k)-labeling problem has been intensively studied in some special cases, i.e. when k=0 (vertex coloring), h=k (vertex coloring the square of the graph) and h=2k (radio- or λ -coloring) but no results are known in the general case for regular grids. In this paper, we completely solve the L(h,k)-labeling problem on regular grids, finding exact values of the span for each value of h and k; only in a small interval we provide different upper and lower bounds.
Combinatorial Gelfand Models
Authors: Adin Ron M. , Postnikov Alex , Roichman Yuval
Published: August 2021

A combinatorial construction of Gelfand models for the symmetric group, for its Iwahori-Hecke algebra and for the hyperoctahedral group is presented.
Skew domino Schensted algorithm and sign-imbalance
Authors: Kim Jang Soo
Published: August 2021

Using growth diagrams, we define a skew domino Schensted algorithm which is a domino analogue of the "Robinson-Schensted algorithm for skew tableaux'' due to Sagan and Stanley. The color-to-spin property of Shimozono and White is extended. As an application, we give a simple generating function for a weighted sum of skew domino tableaux whose special case is a generalization of Stanley's sign-imbalance formula. The generating function gives a method to calculate the generalized sign-imbalance formula.

Nous définissons, à partir de diagrammes de croissances, un algorithme de Schensted pour les dominos gauches. Cet algorithme est un analogue de l'algorithme de Schensted pour les tableaux gauches dû à Sagan et Stanley. Nous généralisons la propriété couleur-à-spin de Shimozono et White. Comme application, nous présentons une fonction génératrice simple pour une somme pondérée de tableaux de dominos gauches qui, dans un cas particulier, généralise la formule de "sign-imbalance'' de Stanley. La fonction génératrice donne aussi lieu à une méthode permettant de calculer la formule de "sign-imbalance''.
New bijective links on planar maps
Authors: Fusy Eric
Published: August 2021

This article describes new bijective links on planar maps, which are of incremental complexity and present original features. The first two bijections $\Phi _{1,2}$ are correspondences on oriented planar maps. They can be considered as variations on the classical edge-poset construction for bipolar orientations on graphs, suitably adapted so as to operate only on the embeddings in a simple local way. In turn, $\Phi_{1,2}$ yield two new bijections $F_{1,2}$ between families of (rooted) maps. (i) By identifying maps with specific constrained orientations, $\Phi_2 \circ \Phi_1$ specialises to a bijection $F_1$ between 2-connected maps and irreducible triangulations; (ii) $F_1$ gives rise to a bijection $F_2$ between loopless maps and triangulations, observing that these decompose respectively into 2-connected maps and into irreducible triangulations in a parallel way.

Cet article décrit de nouveaux liens bijectifs sur les cartes planaires. Nos constructions sont de complexité croissante et présentent des caractéristiques originales. Les deux premières bijections $\Phi _{1,2}$ portent sur des cartes orientées. Elle peuvent être vues comme des variations sur une construction classique de posets sans $N$ à partir d'orientations bipolaires, adaptées ici pour opérer de manière très simple sur le plongement. Les bijections $\Phi _{1,2}$ entrainent à leur tour deux nouvelles bijections $F_{1,2}$ entre familles de cartes (enracinées). (i) En identifiant les cartes avec certaines orientations contraintes, $\Phi_2 \circ \Phi_1$ se spécialise en une bijection $F_1$ entre cartes 2-connexes et triangulations irréductibles, (ii) $F_1$ induit une bijection $F_2$ entre cartes sans boucles et triangulations, qui se décomposent respectivement en cartes 2-connexes et en triangulations irréductibles de manière parallèle.
Bijections for Permutation Tableaux
Authors: Corteel Sylvie , Nadeau Philippe
Published: August 2021

In this paper we propose a new bijection between permutation tableaux and permutations. This bijection shows how natural statistics on the tableaux are equidistributed to classical statistics on permutations: descents, RL-minima and pattern enumerations. We then use the bijection, and a related encoding of tableaux by words, to prove results about the enumeration of permutations with a fixed number of 31-2 patterns, and to define subclasses of permutation tableaux that are in bijection with set partitions. An extended version of this work is available in [6].

Dans cet article nous donnons une bijection entre les tableaux de permutations et les permutations. Cette bijection montre comment des statistiques naturelles sur les tableaux sont équidistribuées avec des statistiques classiques sur les permutations: descentes, minima de droite à gauche et motifs. Cette bijection nous sert ensuite, à l'aide d'un certain codage des tableaux par des mots, à donner des résultats sur l'énumération de permutations avec un nombre fixé de motifs 31-2, et à déterminer certaines sous-classes de tableaux en bijection avec les partitions d'ensembles. Une version étendue de ce travail est disponible [6].
On the link pattern distribution of quarter-turn symmetric FPL configurations
Authors: Duchon Philippe
Published: August 2021

We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures of Razumov and Stroganov and of de Gier. We prove a special case, showing that the link pattern that is conjectured to be the rarest does have the prescribed probability. As a byproduct, we get a formula for the enumeration of a new class of quasi-symmetry of plane partitions.

Nous présentons de nouvelles conjectures portant sur la distribution des schémas de couplage des configurations de boucles compactes (FPL) invariantes, ou presque invariantes, par une rotation d'un quart de tour. Ces nouvelles conjectures étendent des conjectures précédentes dues à Razumov et Stroganov et à de Gier. Dans chaque cas, nous prouvons un cas particulier, en démontrant que le schéma de couplage conjecturé pour être le plus rare a effectivement la probabilité prédite. Nous obtenons également une formule pour l'énumération d'une nouvelle classe de quasi-symétrie de partitions planes.
A survey of multivariate aspects of the contraction method
Authors: Neininger Ralph , Rüschendorf Ludger
Published: August 2021

We survey multivariate limit theorems in the framework of the contraction method for recursive sequences as arising in the analysis of algorithms, random trees or branching processes. We compare and improve various general conditions under which limit laws can be obtained, state related open problems and give applications to the analysis of algorithms and branching recurrences.
Generalized connected domination in graphs
Authors: Kouider Mekkia , Vestergaard Preben Dahl
Published: August 2021

As a generalization of connected domination in a graph G we consider domination by sets having at most k components. The order γ _c^k (G) of such a smallest set we relate to γ _c(G), the order of a smallest connected dominating set. For a tree T we give bounds on γ _c^k (T) in terms of minimum valency and diameter. For trees the inequality γ _c^k (T)≤ n-k-1 is known to hold, we determine the class of trees, for which equality holds.
Counting l-letter subwords in compositions
Authors: Mansour Toufik , Sirhan Basel O.
Published: August 2021

Let ℕ be the set of all positive integers and let A be any ordered subset of ℕ. Recently, Heubach and Mansour enumerated the number of compositions of n with m parts in A that contain the subword τ exactly r times, where τ∈{111,112,221,123}. Our aims are (1) to generalize the above results, i.e., to enumerate the number of compositions of n with m parts in A that contain an ℓ-letter subword, and (2) to analyze the number of compositions of n with m parts that avoid an ℓ-letter pattern, for given ℓ. We use tools such as asymptotic analysis of generating functions leading to Gaussian asymptotic.
An algorithm for generating ribbon tableaux and spin polynomials
Authors: Descouens Francois
Published: August 2021

arXiv:math.CO/0611824

We describe a general algorithm for generating various families of ribbon tableaux and computing their spin polynomials. This algorithm is derived from a new matricial coding. An advantage of this new notation lies in the fact that it permits one to generate ribbon tableaux with skew shapes.
Ambiguity in the m-bonacci numeration system
Authors: Kocábová Petra , Masáková Zuzana , Pelantová Edita
Published: August 2021

Résumé comportant des formules mathématiques, disponible sur le ficher pdf / Abstract with mathematical formulas, available on pdf file.
Foreword to the special issue dedicated to the tenth ''Journées montoises d'informatique théorique''
Authors: Bruyère Véronique , Rigo Michel
Published: August 2021

Held at the Institute of Mathematics of the University of Liège, Liège, September 8―11, 2004

This special issue of Discrete Mathematics & Theoretical Computer Science is dedicated to the tenth "Journées montoises d'informatique théorique" conference (Mons theoretical computer science days) which was held, for the first time, at the Institute of Mathematics of the University of Liège, Belgium, From 8th to 11th September 2004. Previous editions of this conference took place in Mons 1990, 1992, 1994, 1998, in Rouen 1991, in Bordeaux 1993, Marseille 1995, Marne-La-Vallée 2000 and Montpellier 2002.
 This tenth edition can be considered as a widely international one. We were lucky to have almost 85 participants from fourteen different countries: Austria, Belgium, Burkina Faso, Canada, Czech republic, Finland, France, Germany, Israel, Italy, Japan, Norway, Poland and Portugal. The main proportion of researchers participating to this event was coming from France and Italy where a long tradition of combinatorics on words is well established. During four days, 42 contributed talks and 7 invited talks were given, the main topics being combinatorics on words, numeration systems, automata and formal languages theory, coding theory, verification, bio-informatics, number theory, grammars, text algorithms, symbolic dynamics and tilings. The invited speakers were: J. Cassaigne (CNRS, Luminy-Marseille), D. Caucal (IRISIA-CNRS, Rennes), C. Frougny (LIAFA, Université Paris 8), T. Helleseth (University of Bergen), S. Langerman (FNRS, Université Libre de Bruxelles), F. Neven (Limburgs Universitair Centrum, Diepenbeek), M.-F. Sagot (Inria Rhône-Alpes, Université Lyon I).
 We would like to thanks all the participants, the invited speakers and the anonymous referees who made possible this event and special issue. Each paper has been refereed using high scientific standard by two independent referees. Readers of this special issue may wonder why it took so long to obtain it. We have encountered some problems with the formerly chosen journal and for the benefit of the contributors to this issue, we have chosen Discrete Mathematics & Theoretical Computer Science to publish their work.
On the critical exponent of generalized Thue-Morse words
Authors: Massé Alexandre B. , Brlek Srečko , Glen Amy , Labbé Sébastien
Published: August 2021

Automata, Logic and Semantics

For certain generalized Thue-Morse words t, we compute the critical exponent, i.e., the supremum of the set of rational numbers that are exponents of powers in t, and determine exactly the occurrences of powers realizing it.
A geometrical approach of palindromic factors of standard billiard words
Authors: Borel Jean-Pierre
Published: August 2021

Many results are already known, concerning the palindromic factors and the palindomic prefixes of Standard billiard words, i.e., Sturmian words and billiard words in any dimension, starting at the origin. We give new geometrical proofs of these results, especially for the existence in any dimension of Standard billiard words with arbitrary long palindromic prefixes.
Properties of the extremal infinite smooth words
Authors: Brlek Srečko , Melançon Guy , Paquin Geneviève
Published: August 2021

Smooth words are connected to the Kolakoski sequence. We construct the maximal and the minimal in nite smooth words, with respect to the lexicographical order. The naive algorithm generating them is improved by using a reduction of the De Bruijn graph of their factors. We also study their Lyndon factorizations. Finally, we show that the minimal smooth word over the alphabet f1; 3g belongs to the orbit of the Fibonacci word.
Interaction properties of relational periods
Authors: Halava Vesa , Harju Tero , Kärki Tomi
Published: August 2021

Automata, Logic and Semantics

We consider relational periods where the relation is a compatibility relation on words induced by a relation on letters. We introduce three types of periods, namely global, external and local relational periods, and we compare their properties by proving variants of the theorem of Fine and Wilf for these periods.
On symmetric structures of order two
Authors: Bousquet Michel , Lamathe Cédric
Published: August 2021

Combinatorics

Let (w_n)0 < n be the sequence known as Integer Sequence A047749 In this paper, we show that the integer w_n enumerates various kinds of symmetric structures of order two. We ﬁrst consider ternary trees having a reflexive symmetry and we relate all symmetric combinatorial objects by means of bijection. We then generalize the symmetric structures and correspondences to an infinite family of symmetric objects.
A determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata
Authors: Callan David
Published: August 2021

Combinatorics

We show that a determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata. The proof involves a bijection from these automata to certain marked lattice paths and a sign-reversing involution to evaluate the determinant. We also give a formula for the number of acyclic automata with a given set of sources.
The location of the first maximum in the first sojourn of a Dyck path
Authors: Prodinger Helmut
Published: August 2021

Analysis of Algorithms

For Dyck paths (nonnegative symmetric) random walks, the location of the first maximum within the first sojourn is studied. Generating functions and explicit resp. asymptotic expressions for the average are derived. Related parameters are also discussed.
On quadratic residue codes and hyperelliptic curves
Authors: Joyner David
Published: August 2021

Combinatorics

For an odd prime p and each non-empty subset S ⊂ GF(p), consider the hyperelliptic curve X_S deﬁned by y^2 = f_s(x), where f_s(x) = \P_{a2S} (x-a). Using a connection between binary quadratic residue codes and hyperelliptic curves over GF(p), this paper investigates how coding theory bounds give rise to bounds such as the following example: for all sufﬁciently large primes p there exists a subset S ⊂ GF(p) for which the bound |X_S(GF(p))| > 1.39p holds. We also use the quasi-quadratic residue codes deﬁned below to construct an example of a formally self-dual optimal code whose zeta function does not satisfy the "Riemann hypothesis."
Sufficient conditions for labelled 0-1 laws
Authors: Burris Stanley N. , Yeats Karen A.
Published: August 2021

Combinatorics

If F(x) = e^G(x), where F(x) = \Sum f(n)x^n and G(x) = \Sum g(n)x^n, with 0 ≤ g(n) = O(n^θn/n!), θ ∈ (0,1), and gcd(n : g(n) > 0) = 1, then f(n) = o(f(n − 1)). This gives an answer to Compton's request in Question 8.3 [Compton 1987] for an "easily verifiable sufficient condition" to show that an adequate class of structures has a labelled first-order 0-1 law, namely it sufﬁces to show that the labelled component count function is O(n^θn) for some θ ∈ (0,1). It also provides the means to recursively construct an adequate class of structures with a labelled 0-1 law but not an unlabelled 0-1 law, answering Compton's Question 8.4.
Total domination in K₅- and K₆-covered graphs
Authors: Favaron Odile , Karami H. , Sheikholeslami S. M.
Published: August 2021

Graphs and Algorithms

A graph G is Kr-covered if each vertex of G is contained in a Kr-clique. Let $\gamma_t(G)$ denote the total domination number of G. It has been conjectured that every Kr-covered graph of order n with no Kr-component satisﬁes $\gamma_t(G) \le \frac{2n}{r+1}$. We prove that this conjecture is true for r = 5 and 6.
On Compact Encoding of Pagenumber $k$ Graphs
Authors: Gavoille Cyril , Hanusse Nicolas
Published: August 2021

In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;.
Leftmost derivations of propagating scattered context grammars: a new proof
Authors: Masopust Tomáš , Techet Jiří
Published: August 2021

Automata, Logic and Semantics

In 1973, V. Virkkunen proved that propagating scattered context grammars which use leftmost derivations are as powerful as context-sensitive grammars. This paper brings a significantly simplified proof of this result.
Analysis of some parameters for random nodes in priority trees
Authors: Panholzer Alois
Published: August 2021

Analysis of Algorithms

Priority trees are a certain data structure used for priority queue administration. Under the model that all permutations of the numbers 1, . . . , n are equally likely to construct a priority tree of size n we study the following parameters in size-n trees: depth of a random node, number of right edges to a random node, and number of descendants of a random node. For all parameters studied we give limiting distribution results.
Determinants of rational knots
Authors: Kauffman Louis H. , Lopes Pedro
Published: August 2021

Combinatorics

We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd.
Spectral characterizations of sun graphs and broken sun graphs
Authors: Boulet Romain
Published: August 2021

Several matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question ''Which graphs are determined by their spectrum?'' remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by considering some unicyclic graphs. An odd (resp. even) sun is a graph obtained by appending a pendant vertex to each vertex of an odd (resp. even) cycle. A broken sun is a graph obtained by deleting pendant vertices of a sun. In this paper we prove that a sun is determined by its Laplacian spectrum, an odd sun is determined by its adjacency spectrum (counter-examples are given for even suns) and we give some spectral characterizations of broken suns.
Gray codes avoiding matchings
Authors: Dimitrov Darko , Dvořák Tomáš , Gregor Petr , Škrekovski Riste
Published: August 2021

Graphs and Algorithms

A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2(n) binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Q(n), and a cyclic Gray code as a Hamiltonian cycle of Q(n). In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching. Given a matching M and two vertices u, v of Q(n), n >= 4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M, for the existence of a Gray code between u and v that avoids M. As a corollary. we obtain a similar characterization for a cyclic Gray code avoiding M. In particular, in the case that M is a perfect matching, Q(n) has a (cyclic) Gray code that avoids M if and only if Q(n) - M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Q(n) can be extended to a Hamiltonian cycle. Furthermore, our results imply that the problem of Hamilionicity of Q(n) with faulty edges, which is NP-complete in general, becomes polynomial for up to 2(n-1) edges provided they form a matching.
Adjoint functors and tree duality
Authors: Foniok Jan , Tardif Claude
Published: August 2021

Graphs and Algorithms

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.
Ore and Erdős type conditions for long cycles in balanced bipartite graphs
Authors: Adamus Janusz , Adamus Lech
Published: August 2021

Graphs and Algorithms

We conjecture Ore and Erdős type criteria for a balanced bipartite graph of order 2n to contain a long cycle C(2n-2k), where 0 <= k < n/2. For k = 0, these are the classical hamiltonicity criteria of Moon and Moser. The main two results of the paper assert that our conjectures hold for k = 1 as well.
On-line extensible bin packing with unequal bin sizes
Authors: Ye Deshi , Zhang Guochuan
Published: August 2021

Analysis of Algorithms

In the extensible bin packing problem we are asked to pack a set of items into a given number of bins, each with an original size. However, the original bin sizes can be extended if necessary. The goal is to minimize the total size of the bins. We consider the problem with unequal (original) bin sizes and give the complete analysis on a list scheduling algorithm (LS). Namely we present tight bounds of LS for every collection of original bin sizes and every number of bins. We further show better on-line algorithms for the two-bin case and the three-bin case. Interestingly, it is proved that the on-line algorithms have better competitive ratios for unequal bins than for equal bins. Some variants of the problem are also discussed.
Structure of spanning trees on the two-dimensional Sierpinski gasket
Authors: Chang Shu-Chiuan , Chen Lung-Chi
Published: August 2021

Combinatorics

Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage n is a non-negative integer. For any given vertex x of SG(n), we derive rigorously the probability distribution of the degree j ∈{1,2,3,4} at the vertex and its value in the infinite n limit. Adding up such probabilities of all the vertices divided by the number of vertices, we obtain the average probability distribution of the degree j. The corresponding limiting distribution φj gives the average probability that a vertex is connected by 1, 2, 3 or 4 bond(s) among all the spanning tree configurations. They are rational numbers given as φ1=10957/40464, φ2=6626035/13636368, φ3=2943139/13636368, φ4=124895/4545456.
The expected number of inversions after n adjacent transpositions
Authors: Bousquet-Mélou Mireille
Published: August 2021

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.
Non Unitary Random Walks
Authors: Jacquet Philippe
Published: August 2021

Motivated by the recent refutation of information loss paradox in black hole by Hawking, we investigate the new concept of {\it non unitary random walks}. In a non unitary random walk, we consider that the state 0, called the {\it black hole}, has a probability weight that decays exponentially in $e^{-\lambda t}$ for some $\lambda>0$. This decaying probabilities affect the probability weight of the other states, so that the the apparent transition probabilities are affected by a repulsion factor that depends on the factors $\lambda$ and black hole lifetime $t$. If $\lambda$ is large enough, then the resulting transition probabilities correspond to a neutral random walk. We generalize to {\it non unitary gravitational walks} where the transition probabilities are function of the distance to the black hole. We show the surprising result that the black hole remains attractive below a certain distance and becomes repulsive with an exactly reversed random walk beyond this distance. This effect has interesting analogy with so-called dark energy effect in astrophysics.
The Laplacian spread of Cactuses
Authors: Liu Ying
Published: August 2021

Combinatorics

Connected graphs in which any two of its cycles have at most one common vertex are called cactuses. In this paper, we continue the work on Laplacian spread of graphs, and determine the graph with maximal Laplacian spread in all cactuses with n vertices.
Continued fraction expansions for q-tangent and q-cotangent functions
Authors: Prodinger Helmut
Published: August 2021

For 3 different versions of q-tangent resp. q-cotangent functions, we compute the continued fraction expansion explicitly, by guessing the relative quantities and proving the recursive relation afterwards. It is likely that these are the only instances with a ''nice'' expansion. Additional formulae of a similar type are also provided.