bluuebunny/crossref_metadata_2025 huggingface dataset:
Pattern avoidance in dynamical systems
Authors: Amigó José María , Elizalde Sergi , Kennel Matthew B.
Published: August 2021

Orbits generated by discrete-time dynamical systems have some interesting combinatorial properties. In this paper we address the existence of forbidden order patterns when the dynamics is generated by piecewise monotone maps on one-dimensional closed intervals. This means that the points belonging to a sufficiently long orbit cannot appear in any arbitrary order. The admissible patterns are then (the inverses of) those permutations avoiding the so-called forbidden root patterns in consecutive positions. The last part of the paper studies and enumerates forbidden order patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. In spite of their simple structure, shift systems exhibit all important features of low-dimensional chaos, allowing to export the results to other dynamical systems via order-isomorphisms. This paper summarizes some results from [1] and [11].

Les orbites générées par des systèmes dynamiques à temps discret ont quelques propriétés combinatoires intéressantes. Dans cet article on adresse l’existence de motifs d’ordre exclus quand la dynamique est générée par des applications monotones à parts sur des intervalles fermés en une dimension. Ceci signifie que les points appartenant à une orbite suffisamment longue ne peuvent pas apparaître dans un ordre arbitraire. Les motifs admissibles sont alors (les inverses de) ces permutations qui évitent les motifs exclus fondamentaux en positions consécutives. La dernière partie de l’article étudie et énumère les motifs exclus dans les systèmes de déplacement, qui sont des modèles universels dans la théorie de l’information, les systèmes dynamiques et les processus stochastiques. Malgré leur structure simple, les systèmes de déplacement manifestent toutes les propriétés importantes du chaos en basse dimension, permettant exporter les résultats aux autres systèmes dynamiques via des isomorphismes d’ordre. Cet article résume quelques résultats de [1] et [11].
Nested Hilbert Schemes and the nested $q,t$-Catalan series
Authors: Can Mahir Bilen
Published: August 2021

In this paper we study the tangent spaces of the smooth nested Hilbert scheme $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\mathbb{T}^2$-equivariant locally free sheaf on $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients. We call this conjecturally positive polynomial as the "nested $q,t$-Catalan series,'' for it has many conjectural properties similar to that of the $q,t$-Catalan series.

Dans cet article, nous étudions les espaces tangents du schéma de Hilbert emboité lisse $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$ de points du plan, et donnons une formule générale pour le calcul de la caractéristique d’Euler d’un faisceau $\mathbb{T}^2$-équivariant localement libre sur $\mathrm{Hilb}^{n,n-1}(\mathbb{A}^2)$. En appliquant notre resultat a un faisceau particulier, nous conjecturons que le résultat est un polynôme en$q$ et $t$ à coefficents positifs ou nuls. Nous appelons ce polynôme conjecturalement positif la “série de $q; t$-Catalan emboîtée”, car il a de nombreuses propriétés (conjecturées) similaires à celles de la série de $q; t$-Catalan.
Polyominoes determined by involutions
Authors: Disanto Filippo , Rinaldi Simone
Published: August 2021

A permutomino of size n is a polyomino determined by particular pairs $(\pi_1, \pi_2)$ of permutations of length $n$, such that $\pi_1(i) \neq \pi_2(i)$, for $1 \leq i \leq n$. In this paper we consider the class of convex permutominoes which are symmetric with respect to the diagonal $x = y$. We determine the number of these permutominoes according to the dimension and we characterize the class of permutations associated to these objects as particular involutions of length $n$.

Les permutominos de taille $n$ sont des polyominos déterminés par certaines paires de permutations $(\pi_1,\pi_2)$ de taille $n$, telles que $\pi_1(i) \neq \pi_2(i)$, pour tout $1 \leq i \leq n$. Dans cet article nous considérons la classe des permutominos convexes qui sont symétriques par rapport à la diagonale $x=y$. Nous déterminons le nombre de ces permutominos en fonction de leur taille et nous caractérisons la classe des permutations associées à ces objets comme un certain ensemble d'involutions de taille $n$.
Flag enumerations of matroid base polytopes
Authors: Kim Sangwook
Published: August 2021

In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the $\textbf{cd}$-index of a polytope can be expressed when a polytope is cut by a hyperplane, and apply these to the $\textbf{cd}$-index of a matroid base polytope of a rank $2$ matroid.

Dans cet article, nous étudions les structures de drapeau de polytopes de base de matroïde. Nous décrivons des faces de polytopes de base de matroïde en terme des données de matroïde, et donnons des conditions pour les divisions de hyperplane de polytopes de base de matroïde. Nous montrons également comment le $\mathbf{cd}$-index d’un polytope peut être exprimé quand un polytope est coupé par un hyperplane, et appliquons ceux-ci au $\mathbf{cd}$-index d’un polytope de base de matroïde d’un rang $2$ matroïde.
Distances in random Apollonian network structures
Authors: Bodini Olivier , Darrasse Alexis , Soria Michèle
Published: August 2021

In this paper, we study the distribution of distances in random Apollonian network structures (RANS), a family of graphs which has a one-to-one correspondence with planar ternary trees. Using multivariate generating functions that express all information on distances, and singularity analysis for evaluating the coefficients of these functions, we prove a Rayleigh limit distribution for distances to an outermost vertex, and show that the average value of the distance between any pair of vertices in a RANS of order $n$ is asymptotically $\sqrt{n}$.

Nous étudions dans ce papier la distribution des distances dans les structures des réseaux apolloniens aléatoires (RANS), une famille de graphes en bijection avec les arbres ternaires planaires. En s'appuyant sur l'utilisation de séries génératrices multivariées pour décrire toute l'information sur les distances, ainsi que sur l'analyse de singularités pour évaluer les coefficients de ces séries, nous prouvons une distribution limite de Rayleigh pour les distances vers un sommet externe du RANS et montrons que la distance moyenne entre deux sommets quelconques d'un RANS d'ordre $n$ est asymptotiquement $\sqrt{n}$.
From Bruhat intervals to intersection lattices and a conjecture of Postnikov
Authors: Hultman Axel , Linusson Svante , Shareshian John , Sjöstrand Jonas
Published: August 2021

We prove the conjecture of A. Postnikov that ($\mathrm{A}$) the number of regions in the inversion hyperplane arrangement associated with a permutation $w \in \mathfrak{S}_n$ is at most the number of elements below $w$ in the Bruhat order, and ($\mathrm{B}$) that equality holds if and only if $w$ avoids the patterns $4231$, $35142$, $42513$ and $351624$. Furthermore, assertion ($\mathrm{A}$) is extended to all finite reflection groups.

Nous prouvons la conjecture de A. Postnikov que ($\mathrm{A}$) le nombre de régions dans l'arrangement d'hyperplans inverses associés à la permutation $w \in \mathfrak{S}_n$ est au plus égal au nombre d'éléments en dessous de $w$ dans l'ordre de Bruhat, et ($\mathrm{B}$) il y a égalité si et seulement si $w$ évite les motifs $4231$, $35142$, $42513$ et $351624$. De plus, l'affirmation ($\mathrm{A}$) est généralisée à tous les groupes de réflexion finis.
$(\ell, 0)$-Carter Partitions and their crystal theoretic interpretation
Authors: Berg Chris , Vazirani Monica
Published: August 2021

In this paper we give an alternate combinatorial description of the "$(\ell,0)$-Carter partitions''. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas ($\textit{A q-analogue of the Jantzen-Schaper theorem}$). The condition of being an $(\ell,0)$-Carter partition is fundamentally related to the hook lengths of the partition. The representation-theoretic significance of their combinatoric on an $\ell$-regular partition is that it indicates the irreducibility of the corresponding Specht module over the finite Hecke algebra. We use our result to find a generating series which counts the number of such partitions, with respect to the statistic of a partition's first part. We then apply our description of these partitions to the crystal graph $B(\Lambda_0)$ of the basic representation of $\widehat{\mathfrak{sl}_{\ell}}$, whose nodes are labeled by $\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $(\ell,0)$-Carter partitions in the graph of $B(\Lambda_0)$.

Dans cet article, nous donnons une description combinatoire alternative des partitions "$(\ell,0)$-Carter". Notre théorème principal est une équivalence entre notre combinatoire et celle introduite par James et Mathas ($\textit{A q-analogue of the Jantzen-Schaper theorem}$). La propriété $(\ell,0)$-Carter est fondamentalement liée aux longueurs des équerres de la partition. En terme de théorie des représentations, leur combinatoire pour une partition $\ell$-régulière permet de déterminer l'irréducibilité du module de Specht spécialisé sur l’algèbre de Hecke finie. Nous utilisons notre résultat pour déterminer leur série génératrice en fonction de la taille de la première part. Nous utilisons ensuite notre description de ces partitions au graphe cristallin $B(\Lambda _0)$ de la représentation basique de $\widehat{\mathfrak{sl}_{\ell}}$, dont les nœuds sont étiquetés par les partitions $\ell$-régulières. Nous donnons une règle cristalline relativement simple permettant d'engendrer toutes les partitions $\ell$-régulières $(\ell,0)$-Carter dans le graphe de $B(\Lambda _0)$.
Algebraic shifting and strongly edge decomposable complexes
Authors: Murai Satoshi
Published: August 2021

Let $\Gamma$ be a simplicial complex with $n$ vertices, and let $\Delta (\Gamma)$ be either its exterior algebraic shifted complex or its symmetric algebraic shifted complex. If $\Gamma$ is a simplicial sphere, then it is known that (a) $\Delta (\Gamma)$ is pure and (b) $h$-vector of $\Gamma$ is symmetric. Kalai and Sarkaria conjectured that if $\Gamma$ is a simplicial sphere then its algebraic shifting also satisfies (c) $\Delta (\Gamma) \subset \Delta (C(n,d))$, where $C(n,d)$ is the boundary complex of the cyclic $d$-polytope with $n$ vertices. We show this conjecture for strongly edge decomposable spheres introduced by Nevo. We also show that any shifted simplicial complex satisfying (a), (b) and (c) is the algebraic shifted complex of some simplicial sphere.
On Sampling Colorings of Bipartite Graphs
Authors: Balasubramanian R. , Subramanian C.R.
Published: August 2021

We study the problem of efficiently sampling k-colorings of bipartite graphs. We show that a class of markov chains cannot be used as efficient samplers. Precisely, we show that, for any k, 6 ≤ k ≤ n^\1/3-ε \, ε > 0 fixed, \emphalmost every bipartite graph on n+n vertices is such that the mixing time of any markov chain asymptotically uniform on its k-colorings is exponential in n/k^2 (if it is allowed to only change the colors of O(n/k) vertices in a single transition step). This kind of exponential time mixing is called \emphtorpid mixing. As a corollary, we show that there are (for every n) bipartite graphs on 2n vertices with Δ (G) = Ω (\ln n) such that for every k, 6 ≤ k ≤ Δ /(6 \ln Δ ), each member of a large class of chains mixes torpidly. While, for fixed k, such negative results are implied by the work of CDF, our results are more general in that they allow k to grow with n. We also show that these negative results hold true for H-colorings of bipartite graphs provided H contains a spanning complete bipartite subgraph. We also present explicit examples of colorings (k-colorings or H-colorings) which admit 1-cautious chains that are ergodic and are shown to have exponential mixing time. While, for fixed k or fixed H, such negative results are implied by the work of CDF, our results are more general in that they allow k or H to vary with n.
Coding partitions
Authors: Burderi Fabio , Restivo Antonio
Published: August 2021

Motivated by the study of decipherability conditions for codes weaker than Unique Decipherability (UD), we introduce the notion of coding partition. Such a notion generalizes that of UD code and, for codes that are not UD, allows to recover the ''unique decipherability" at the level of the classes of the partition. By tacking into account the natural order between the partitions, we define the characteristic partition of a code X as the finest coding partition of X. This leads to introduce the canonical decomposition of a code in at most one unambiguous component and other (if any) totally ambiguous components. In the case the code is finite, we give an algorithm for computing its canonical partition. This, in particular, allows to decide whether a given partition of a finite code X is a coding partition. This last problem is then approached in the case the code is a rational set. We prove its decidability under the hypothesis that the partition contains a finite number of classes and each class is a rational set. Moreover we conjecture that the canonical partition satisfies such a hypothesis. Finally we consider also some relationships between coding partitions and varieties of codes.
Independent sets in graphs with an excluded clique minor
Authors: Wood David R.
Published: August 2021

Graphs and Algorithms

Let G be a graph with n vertices, with independence number α, and with no Kt+1-minor for some t ≥ 5. It is proved that (2α - 1)(2t - 5) ≥ 2n - 5. This improves upon the previous best bound whenever n≥2/5t2.
On the tileability of polygons with colored dominoes
Authors: Worman Chris , Yang Boting
Published: August 2021

Analysis of Algorithms

We consider questions concerning the tileability of orthogonal polygons with colored dominoes. A colored domino is a rotatable 2 × 1 rectangle that is partitioned into two unit squares, which are called faces, each of which is assigned a color. In a colored domino tiling of an orthogonal polygon P, a set of dominoes completely covers P such that no dominoes overlap and so that adjacent faces have the same color. We demonstrated that for simple layout polygons that can be tiled with colored dominoes, two colors are always sufficient. We also show that for tileable non-simple layout polygons, four colors are always sufficient and sometimes necessary. We describe an O(n) time algorithm for computing a colored domino tiling of a simple orthogonal polygon, if such a tiling exists, where n is the number of dominoes used in the tiling. We also show that deciding whether or not a non-simple orthogonal polygon can be tiled with colored dominoes is NP-complete.
Arithmetics in β-numeration
Authors: Bernat Julien
Published: August 2021

Analysis of Algorithms

The β-numeration, born with the works of Rényi and Parry, provides a generalization of the notions of integers, decimal numbers and rational numbers by expanding real numbers in base β, where β>1 is not an integer. One of the main differences with the case of numeration in integral base is that the sets which play the role of integers, decimal numbers and rational numbers in base β are not stable under addition or multiplication. In particular, a fractional part may appear when one adds or multiplies two integers in base β. When β is a Pisot number, which corresponds to the most studied case, the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β are bounded by constants which only depend on β. We prove that, for any Perron number β, the set of finite or ultimately periodic fractional parts of the sum, or the product, of two integers in base β is finite. Additionally, we prove that it is possible to compute this set for the case of addition when β is a Parry number. As a consequence, we deduce that, when β is a Perron number, there exist bounds, which only depend on β, for the lengths of the finite fractional parts that may appear when one adds or multiplies two integers in base β. Moreover, when β is a Parry number, the bound associated with the case of addition can be explicitly computed.
A lower bound for approximating the Grundy number
Authors: Kortsarz Guy
Published: August 2021

Graphs and Algorithms

The grundy numbering of a graph is the maximum number of colors used by on-line first-fit coloring, under the worst order of arrival of vertices. The grundy numbering problem is to find this ordering. We prove that there is a constant c>1 so that approximating the grundy numbering problem within c is not possible, unless NP ⊆ RP
Gray code order for Lyndon words
Authors: Vajnovszki Vincent
Published: August 2021

At the 4th Conference on Combinatorics on Words, Christophe Reutenauer posed the question of whether the dual reflected order yields a Gray code on the Lyndon family. In this paper we give a positive answer. More precisely, we present an O(1)-average-time algorithm for generating length n binary pre-necklaces, necklaces and Lyndon words in Gray code order.
The \v Cerný conjecture for aperiodic automata
Authors: Trahtman A. N.
Published: August 2021

A word w is called a synchronizing (recurrent, reset, directable) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some specific state; a DFA that has a synchronizing word is said to be synchronizable. Cerny conjectured in 1964 that every n-state synchronizable DFA possesses a synchronizing word of length at most (n-1)2. We consider automata with aperiodic transition monoid (such automata are called aperiodic). We show that every synchronizable n-state aperiodic DFA has a synchronizing word of length at most n(n-1)/2. Thus, for aperiodic automata as well as for automata accepting only star-free languages, the Cerny conjecture holds true.
Tribes of cubic partial cubes
Authors: Klavžar Sandi , Shpectorov Sergey
Published: August 2021

Graphs and Algorithms

Partial cubes are graphs isometrically embeddable into hypercubes. Three infinite families and a few sporadic examples of cubic partial cubes are known. The concept of a tribe is introduced as means to systematize the known examples and establish relations among them. Efficient methods of computation of tribes are developed and several concrete tribes, that include known, as well as new cubic partial cubes, are computed by hand and with the use of a computer.
Asymptotic behaviour of a non-commutative rational series with a nonnegative linear representation
Authors: Dumas Philippe , Lipmaa Helger , Wallén Johan
Published: August 2021

Analysis of Algorithms

We analyse the asymptotic behaviour in the mean of a non-commutative rational series, which originates from differential cryptanalysis, using tools from probability theory, and from analytic number theory. We derive a Fourier representation of a first-order summation function obtained by interpreting this rational series as a non-classical rational sequence via the octal numeration system. The method is applicable to a wide class of sequences rational with respect to a numeration system essentially under the condition that they admit a linear representation with nonnegative coefficients.
Latin square Thue-Morse sequences are overlap-free
Authors: Tompkins C. Robinson
Published: August 2021

Automata, Logic and Semantics

We define a morphism based upon a Latin square that generalizes the Thue-Morse morphism. We prove that fixed points of this morphism are overlap-free sequences, generalizing results of Allouche - Shallit and Frid.
"Trivializing'' generalizations of some Izergin-Korepin-type determinants
Authors: Amdeberhan Tewodros , Zeilberger Doron
Published: August 2021

Combinatorics

We generalize (and hence trivialize and routinize) numerous explicit evaluations of determinants and pfaffians due to Kuperberg, as well as a determinant of Tsuchiya. The level of generality of our statements render their proofs easy and routine, by using Dodgson Condensation and/or Krattenthaler's factor exhaustion method.
FP/FIFO scheduling: coexistence of deterministic and probabilistic QoS guarantees
Authors: Minet Pascale , Martin Steven , Saidane Leila Azouz , Azzaz Skander
Published: August 2021

Distributed Computing and Networking

In this paper, we focus on applications having quantitative QoS (Quality of Service) requirements on their end-to-end response time (or jitter). We propose a solution allowing the coexistence of two types of quantitative QoS garantees, deterministic and probabilistic, while providing a high resource utilization. Our solution combines the advantages of the deterministic approach and the probabilistic one. The deterministic approach is based on a worst case analysis. The probabilistic approach uses a mathematical model to obtain the probability that the response time exceeds a given value. We assume that flows are scheduled according to non-preemptive FP/FIFO. The packet with the highest fixed priority is scheduled first. If two packets share the same priority, the packet arrived first is scheduled first. We make no particular assumption concerning the flow priority and the nature of the QoS guarantee requested by the flow. An admission control derived from these results is then proposed, allowing each flow to receive a quantitative QoS guarantee adapted to its QoS requirements. An example illustrates the merits of the coexistence of deterministic and probabilistic QoS guarantees.
Exponential bounds and tails for additive random recursive sequences
Authors: Rüschendorf Ludger , Schopp Eva-Maria
Published: August 2021

Analysis of Algorithms

Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method. The paper generalizes results of Rösler (% \citeyearNPRoesler:91, % \citeyearNPRoesler:92) and % \citeNNeininger:05 on subgaussian tails to more general classes of additive random recursive sequences. It also gives sufficient conditions for tail bounds of the form \exp(-a t^p) which are based on a characterization of \citeNKasahara:78.
On morphisms preserving infinite Lyndon words
Authors: Richomme Gwenael
Published: August 2021

In a previous paper, we characterized free monoid morphisms preserving finite Lyndon words. In particular, we proved that such a morphism preserves the order on finite words. Here we study morphisms preserving infinite Lyndon words and morphisms preserving the order on infinite words. We characterize them and show relations with morphisms preserving Lyndon words or the order on finite words. We also briefly study morphisms preserving border-free words and those preserving the radix order.
Strong chromatic index of products of graphs
Authors: Togni Olivier
Published: August 2021

Graphs and Algorithms

The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for strong chromatic index of three different products of graphs in term of the strong chromatic index of each factor. For the cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes.
Strong Oriented Chromatic Number of Planar Graphs without Short Cycles
Authors: Montassier Mickael , Ochem Pascal , Pinlou Alexandre
Published: August 2021

Let M be an additive abelian group. An M-strong-oriented coloring of an oriented graph G is a mapping f from V(G) to M such that f(u) <> j(v) whenever uv is an arc in G and f(v)−f(u) <> −(f(t)−f(z)) whenever uv and zt are two arcs in G. The strong oriented chromatic number of an oriented graph is the minimal order of a group M such that G has an M-strong-oriented coloring. This notion was introduced by Nesetril and Raspaud [Ann. Inst. Fourier, 49(3):1037-1056, 1999]. We prove that the strong oriented chromatic number of oriented planar graphs without cycles of lengths 4 to 12 (resp. 4 or 6) is at most 7 (resp. 19). Moreover, for all i ≥ 4, we construct outerplanar graphs without cycles of lengths 4 to i whose oriented chromatic number is 7.
Shifts with decidable language and non-computable entropy
Authors: Hertling Peter , Spandl Christoph
Published: August 2021

Automata, Logic and Semantics

We consider subshifts of the full shift of all binary bi-infinite sequences. On the one hand, the topological entropy of any subshift with computably co-enumerable language is a right-computable real number between 0 and 1. We show that, on the other hand, any right-computable real number between 0 and 1, whether computable or not, is the entropy of some subshift with even polynomial time decidable language. In addition, we show that computability of the entropy of a subshift does not imply any kind of computability of the language of the subshift
Progress on the traceability conjecture for oriented graphs
Authors: Frick Marietjie , Katrenič Peter
Published: August 2021

Graphs and Algorithms

A digraph is k-traceable if each of its induced subdigraphs of order k is traceable. The Traceability Conjecture is that for k ≥ 2 every k-traceable oriented graph of order at least 2k − 1 is traceable. The conjecture has been proved for k ≤ 5. We prove that it also holds for k = 6.
A note on compact and compact circular edge-colorings of graphs
Authors: Dereniowski Dariusz , Nadolski Adam
Published: August 2021

Graphs and Algorithms

We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.
Counting descents, rises, and levels, with prescribed first element, in words
Authors: Kitaev Sergey , Mansour Toufik , Remmel Jeff
Published: August 2021

Combinatorics

Recently, Kitaev and Remmel refined the well-known permutation statistic "descent" by fixing parity of one of the descent's numbers which was extended and generalized in several ways in the literature. In this paper, we shall fix a set partition of the natural numbers N,(N1, ..., Ns), and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = 1, ..., k. In particular, we refine and generalize some of the results by Burstein and Mansour
Convergence of some leader election algorithms
Authors: Janson Svante , Lavault Christian , Louchard Guy
Published: August 2021

We start with a set of $n$ players. With some probability $P(n,k)$, we kill $n-k$ players; the other ones stay alive, and we repeat with them. What is the distribution of the number $X_n$ of \emph{phases} (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions $P(n,k)$, including stochastic monotonicity and the assumption that roughly a fixed proportion $\al$ of the players survive in each round. We prove a kind of convergence in distribution for $X_n - \log_{1/\!\alpha}(n)$; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable $Z$ such that $d\l(X_n, \lceil Z + \log_{1/\!\alpha} (n)\rceil\r) \to 0$, where $d$ is either the total variation distance or the Wasserstein distance. Applications of the general result include the leader election algorithm where players are eliminated by independent coin tosses and a variation of the leader election algorithm proposed by W.R. Franklin. We study the latter algorithm further, including numerical results.
Waiting Time Distribution for Pattern Occurrence in a Constrained Sequence: an Embedding Markov Chain Approach
Authors: Nuel Gregory
Published: August 2021

Analysis of Algorithms

In this paper we consider the distribution of a pattern of interest in a binary random (d; k)-sequence generated by a Markov source. Such constrained sequences are frequently encountered in communication systems. Unlike the previous approach based on generating function we have chosen here to use Markov chain embedding techniques. By doing so, we get both previous results (sequence constrained up to the rth occurrence), and new ones (sequence constrained up to its end). We also provide in both cases efficient algorithms using basic linear algebra only. We then compare our numerical results to previous ones and finally propose several interesting extensions of our method which further illustrate the usefulness of our approach. That is to say higher order Markov chains, renewal occurrences rather than overlapping ones, heterogeneous models, more complex patterns of interest, and multistate trial sequences.
Diophantine Approximation, Ostrowski Numeration and the Double-Base Number System
Authors: Berthe Valerie , Imbert Laurent
Published: August 2021

Analysis of Algorithms

A partition of $x > 0$ of the form $x = \sum_i 2^{a_i}3^{b_i}$ with distinct parts is called a double-base expansion of $x$. Such a representation can be obtained using a greedy approach, assuming one can efficiently compute the largest \mbox{$\{2,3\}$-integer}, i.e., a number of the form $2^a3^b$, less than or equal to $x$. In order to solve this problem, we propose an algorithm based on continued fractions in the vein of the Ostrowski number system, we prove its correctness and we analyse its complexity. In a second part, we present some experimental results on the length of double-base expansions when only a few iterations of our algorithm are performed.
Edge condition for long cycles in bipartite graphs
Authors: Adamus Lech
Published: August 2021

Graphs and Algorithms

The following problem was solved by Woodall in 1972: for any pair of nonnegative integers n and k < n/2 - 1 find the minimum integer g(n, k) such that every graph with n vertices and at least g(n, k) edges contains a cycle of length n - k. Woodall proved even more: the size g(n, k), in fact, guarantees the existence of cycles C, for all 3 <= p <= n - k.

 In the paper an analogous problem for bipartite graphs is considered. It is proved that every bipartite graph with color classes of cardinalities m and n, m <= n, and size greater than n(m - k - 1) + k + 1 contains a cycle of length 2m - 2k, where m >= 1/2k(2) + 3/2k + 4, k is an element of N. The bound on the number of edges is best possible. Moreover, this size condition guarantees the existence of cycles of all even lengths up to 2m - 2k. We also characterize all extremal graphs for this problem. Finally, we conjecture that the condition on the order may be relaxed to m >= 2k + 2.
On the chromatic number of some flip graphs
Authors: Fabila-Monroy Ruy , Flores-Peñaloza David , Huemer Clemens , Hurtado Ferran , Urrutia Jorge , Wood David R.
Published: August 2021

Graphs and Algorithms

This paper studies the chromatic number of the following four flip graphs (under suitable definitions of a flip): the flip graph of perfect matchings of a complete graph of even order, the flip graph of triangulations of a convex polygon (the associahedron), the flip graph of non-crossing Hamiltonian paths of a set of points in convex position, and the flip graph of triangles in a convex point set. We give tight bounds for the latter two cases and upper bounds for the first two.
String pattern avoidance in generalized non-crossing trees
Authors: Sun Yidong , Wang Zhiping
Published: August 2021

Combinatorics

The problem of string pattern avoidance in generalized non-crossing trees is studied. The generating functions for generalized non-crossing trees avoiding string patterns of length one and two are obtained. The Lagrange inversion formula is used to obtain the explicit formulas for some special cases. A bijection is also established between generalized non-crossing trees with special string pattern avoidance and little Schr ̈oder paths.
Long cycles in hypercubes with distant faulty vertices
Authors: Gregor Petr , Škrekovski Riste
Published: August 2021

Graphs and Algorithms

In this paper, we study long cycles in induced subgraphs of hypercubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced subgraph of Q(n) with minimum degree n - 1 contains a cycle of length at least 2(n) - 2(f) where f is the number of removed vertices. This length is the best possible when all removed vertices are from the same bipartite class of Q(n). Next, we prove that every induced subgraph of Q(n) obtained by removing vertices of some given set M of edges of Q(n) contains a Hamiltonian cycle if every two edges of M are at distance at least 3. The last result shows that the shell of every linear code with odd minimum distance at least 3 contains a Hamiltonian cycle. In all these results we obtain significantly more tolerable faulty vertices than in the previously known results. We also conjecture that every induced subgraph of Q(n) obtained by removing a balanced set of vertices with minimum distance at least 3 contains a Hamiltonian cycle.
Enumeration of words by the sum of differences between adjacent letters
Authors: Mansour Toufik
Published: August 2021

Combinatorics

We consider the sum u of differences between adjacent letters of a word of n letters, chosen uniformly at random from a given alphabet. This paper obtains the enumerating generating function for the number of such words with respect to the sum u, as well as explicit formulas for the mean and variance of u.
The Eulerian distribution on centrosymmetric involutions
Authors: Barnabei Marilena , Bonetti Flavio , Silimbani Matteo
Published: August 2021

Combinatorics

We present an extensive study of the Eulerian distribution on the set of centrosymmetric involutions, namely, involutions in S(n) satisfying the property sigma(i) + sigma(n + 1 - i) = n + 1 for every i = 1 ... n. We find some combinatorial properties for the generating polynomial of such distribution, together with an explicit formula for its coefficients. Afterwards, we carry out an analogous study for the subset of centrosymmetric involutions without fixed points.
On the Number of Balanced Words of Given Length and Height over a Two-Letter Alphabet
Authors: Bédaride Nicolas , Domenjoud Eric , Jamet Damien , Rémy Jean-Luc
Published: August 2021

We exhibit a recurrence on the number of discrete line segments joining two integer points in the plane using an encoding of such segments as balanced words of given length and height over the two-letter alphabet $\{0,1\}$. We give generating functions and study the asymptotic behaviour. As a particular case, we focus on the symmetrical discrete segments which are encoded by balanced palindromes.
Binary Labelings for Plane Quadrangulations and their Relatives
Authors: Felsner Stefan , Huemer Clemens , Kappes Sarah , Orden David
Published: August 2021

Graphs and Algorithms

Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder's one for triangulations: Apart from being in bijection with tree decompositions, paths in these trees allow to define the regions of a vertex such that counting faces in them yields an algorithm for embedding the quadrangulation, in this case on a 2-book. Furthermore, as Schnyder labelings have been extended to 3-connected plane graphs, we are able to extend our labeling from quadrangulations to a larger class of 2-connected bipartite graphs.
Acyclic colourings of graphs with bounded degree
Authors: Borowiecki Mieczyslaw , Fiedorowicz Anna , Jesse-Jozefczyk Katarzyna , Sidorowicz Elzbieta
Published: August 2021

Graphs and Algorithms

A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgraph induced in G by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, there are no bichromatic alternating cycles. In 1999 Boiron et al. conjectured that a graph G with maximum degree at most 3 has an acyclic 2-colouring such that the set of vertices in each colour induces a subgraph with maximum degree at most 2. In this paper we prove this conjecture and show that such a colouring of a cubic graph can be determined in polynomial time. We also prove that it is an NP-complete problem to decide if a graph with maximum degree 4 has the above mentioned colouring.
Split-critical and uniquely split-colorable graphs
Authors: Ekim Tınaz , Ries Bernard , De Werra Dominique
Published: August 2021

Graphs and Algorithms

The split-coloring problem is a generalized vertex coloring problem where we partition the vertices into a minimum number of split graphs. In this paper, we study some notions which are extensively studied for the usual vertex coloring and the cocoloring problem from the point of view of split-coloring, such as criticality and the uniqueness of the minimum split-coloring. We discuss some properties of split-critical and uniquely split-colorable graphs. We describe constructions of such graphs with some additional properties. We also study the effect of the addition and the removal of some edge sets on the value of the split-chromatic number. All these results are compared with their cochromatic counterparts. We conclude with several research directions on the topic.
The absence of a pattern and the occurrences of another
Authors: Bóna Miklós
Published: August 2021

Following a question of J. Cooper, we study the expected number of occurrences of a given permutation pattern q in permutations that avoid another given pattern r. In some cases, we find the pattern that occurs least often, (resp. most often) in all r-avoiding permutations. We also prove a few exact enumeration formulae, some of which are surprising.
Some properties of semiregular cages
Authors: Balbuena Camino , Marcote Xavier , Gonzalez-Moreno Diego
Published: August 2021

Graphs and Algorithms

A graph with degree set \r, r + 1\ is said to be semiregular. A semiregular cage is a semiregular graph with given girth g and the least possible order. First, an upper bound on the diameter of semiregular graphs with girth g and order close enough to the minimum possible value is given in this work. As a consequence, these graphs are proved to be maximally connected when the girth g >= 7 is odd. Moreover an upper bound for the order of semiregular cages is given and, as an application, every semiregular cage with degree set \r, r + 1\ is proved to be maximally connected for g is an element of \6, 8\, and when g = 12 for r >= 7 and r not equal 20. Finally it is also shown that every (\r, r + 1\; g)-cage is 3-connected.
Lower Bounds on the Area Requirements of Series-Parallel Graphs
Authors: Frati Fabrizio
Published: August 2021

Graphs and Algorithms

We show that there exist series-parallel graphs requiring Omega(n2(root log n)) area in any straight-line or poly-line grid drawing. Such a result is achieved in two steps. First, we show that, in any straight-line or poly-line drawing of K(2,n), one side of the bounding box has length Omega(n), thus answering two questions posed by Biedl et al. Second, we show a family of series-parallel graphs requiring Omega(2(root log n)) width and Omega(2(root log n)) height in any straight-line or poly-line grid drawing. Combining the two results, the Omega(n2(root log n)) area lower bound is achieved.
Deciding whether the ordering is necessary in a Presburger formula
Authors: Choffrut Christian , Frigeri Achille
Published: August 2021

Automata, Logic and Semantics

We characterize the relations which are first-order definable in the model of the group of integers with the constant 1. This allows us to show that given a relation defined by a first-order formula in this model enriched with the usual ordering, it is recursively decidable whether or not it is first-order definable without the ordering.
A divergent generating function that can be summed and analysed analytically
Authors: Janson Svante
Published: August 2021

We study a recurrence relation, originating in combinatorial problems, where the generating function, as a formal power series, satisfies a differential equation that can be solved in a suitable domain; this yields an analytic function in a domain, but the solution is singular at the origin and the generating function has radius of convergence 0. Nevertheless, the solution to the recurrence can be obtained from the analytic solution by finding an asymptotic series expansion. Conversely, the analytic solution can be obtained by summing the generating function by the Borel summation method. This is an explicit example, which we study detail, of a behaviour known to be typical for a large class of holonomic functions. We also express the solution using Bessel functions and Lommel polynomials.
On the support of the free Lie algebra: the Schutzenberger problems
Authors: Michos Ioannis C.
Published: August 2021

Combinatorics

M.-P. Schutzenberger asked to determine the support of the free Lie algebra L(Zm) (A) on a finite alphabet A over the ring Z(m) of integers mod m and all pairs of twin and anti-twin words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We characterize the complement of the support of L(Zm) (A) in A* as the set of all words w such that m divides all the coefficients appearing in the monomials of l* (w), where l* is the adjoint endomorphism of the left normed Lie bracketing l of the free Lie ring. Calculating l* (w) via the shuffle product, we recover the well known result of Duchamp and Thibon (Discrete Math. 76 (1989) 123-132) for the support of the free Lie ring in a much more natural way. We conjecture that two words u and v of common length n, which lie in the support of the free Lie ring, are twin (resp. anti-twin) if and only if either u = v or n is odd and u = (v) over tilde (resp. if n is even and u = (v) over tilde), where (v) over tilde denotes the reversal of v and we prove that it suffices to show this for a two-lettered alphabet. These problems can be rephrased, for words of length n, in terms of the action of the Dynkin operator l(n) on lambda-tabloids, where lambda is a partition of n. Representing a word w in two letters by the subset I of [n] = \1, 2, ... , n\ that consists of all positions that one of the letters occurs in w, the computation of l* (w) leads us to the notion of the Pascal descent polynomial p(n)(I), a particular commutative multi-linear polynomial which is equal to the signed binomial coefficient when vertical bar I vertical bar = 1. We provide a recursion formula for p(n) (I) and show that if m inverted iota Sigma(i is an element of I)(1)(i-1) (n - 1 i - 1), then w lies in the support of L(Zm) (A).
Combinatorics, Groups, Algorithms, and Complexity: Conference in honor of Laci Babai's 60th birthday
Authors: Seress Akos , Szegedy Mario
Published: August 2021

Honoring László (Laci) Babai's 60th birthday, the conference "Combinatorics, Groups, Algorithms, and Complexity" (Ohio State University, March 15-25, 2010) explored the links between the areas mentioned in the title. These areas represent Laci's wide interests in mathematics and theoretical computer science; his work has revealed and enriched many of the interconnections between them. The conference had 109 participants from North America, Europe, Asia, and Australia (31 of them from overseas), including 3 Nevanlinna prize winners, 32 students, 13 postdocs, 20 females, and 18 former and current students of Laci Babai. The program consisted of 73 talks and a problem session. The full list of talks can be found in the introductory article by the guest editors of this special issue who also served as the organizers of the conference. We thank all participants and speakers for the success of the conference. We wish to express our gratitude to the National Science Foundation, National Security Agency, and The Ohio State Mathematical Research Institute for their generous support. This special issue contains papers in the conference topics, but not necessarily coinciding with the authors' talks at the conference. Each paper has been peer-reviewed. Toniann Pitassi, László Pyber, Uwe Schöning, Jiří Sgall, and Aner Shalev served with us as editors of this special issue. We thank for their work as well as for the assistance of the anonymous referees.
Nordhaus-Gaddum Type Results for Total Domination
Authors: Henning Michael , Joubert Ernst , Southey Justin
Published: August 2021

Graph Theory

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper we study Nordhaus-Gaddum-type results for total domination. We examine the sum and product of γt(G1) and γt(G2) where G1 ⊕G2 = K(s,s), and γt is the total domination number. We show that the maximum value of the sum of the total domination numbers of G1 and G2 is 2s+4, with equality if and only if G1 = sK2 or G2 = sK2, while the maximum value of the product of the total domination numbers of G1 and G2 is max{8s,⌊(s+6)2/4 ⌋}.
Cycle transversals in bounded degree graphs
Authors: Groshaus Marina , Hell Pavol , Klein Sulamita , Nogueira Loana Tito , Protti Fábio
Published: August 2021

Graphs and Algorithms

In this work we investigate the algorithmic complexity of computing a minimum C(k)-transversal, i.e., a subset of vertices that intersects all the chordless cycles with k vertices of the input graph, for a fixed k \textgreater= 3. For graphs of maximum degree at most three, we prove that this problem is polynomial-time solvable when k \textless= 4, and NP-hard otherwise. For graphs of maximum degree at most four, we prove that this problem is NP-hard for any fixed k \textgreater= 3. We also discuss polynomial-time approximation algorithms for computing C(3)-transversals in graphs of maximum degree at most four, based on a new decomposition theorem for such graphs that leads to useful reduction rules.
The largest singletons in weighted set partitions and its applications
Authors: Sun Yidong , Xu Yanjie
Published: August 2021

Combinatorics

Recently, Deutsch and Elizalde studied the largest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let A (n,k) (t) denote the total weight of partitions on [n + 1] = \1,2,..., n + 1\ with the largest singleton \k + 1\. In this paper, explicit formulas for A (n,k) (t) and many combinatorial identities involving A (n,k) (t) are obtained by umbral operators and combinatorial methods. In particular, the permutation case leads to an identity related to tree enumerations, namely, [GRAPHICS] where D-k is the number of permutations of [k] with no fixed points.