bluuebunny/crossref_metadata_2025 huggingface dataset:
$Star^1$-convex functions on tropical linear spaces of complete graphs
Authors: Escobar Laura
Published: August 2021

Given a fan $\Delta$ and a cone $\sigma \in \Delta$ let $star^1(\sigma )$ be the set of cones that contain $\sigma$ and are one dimension bigger than $\sigma$ . In this paper we study two cones of piecewise linear functions defined on $\delta$ : the cone of functions which are convex on $star^1(σ\sigma)$ for all cones, and the cone of functions which are convex on $star^1(σ\sigma)$ for all cones of codimension 1. We give nice combinatorial descriptions for these two cones given two different fan structures on the tropical linear space of complete graphs. For the complete graph $K_5$, we prove that with the finer fan subdivision the two cones are not equal, but with the coarser subdivision they are the same. This gives a negative answer to a question of Gibney-Maclagan that for the finer subdivision the two cones are the same.

Soit $\Delta$ un fan, pour $\sigma \in \Delta$ nous définissons $star^1(\sigma )$ comme l'ensemble de cônes qui contiennent $\sigma$ dont la dimension est un de plus que la dimension de $\sigma$ . Nous étudions deux cônes d'applications linéaires par morceaux définis sur $\Delta$ : le cône de fonctions convexes sur$star^1(\sigma )$, où $\sigma \in \Delta$ est un cône quelconque, et le cône de fonctions convexes sur $star^1(σ\sigma)$ où σ est un cône de codimension 1. étant donnés deux structures sur l'espace tropical linéaire de graphes complets, nous donnons de beaux descriptions combinatoires des cônes décrits en haut. Pour le graphe complet $K_5$, on démontre que avec la subdivision en fans plus fine, les deux cônes sont différentes, mais avec la subdivision plus gros ils sont cônes sont les mêmes. Ce résultant réponde négativement une question de Gibney-Maclagan.
Fusion coefficients
Authors: Morse Jennifer , Schilling Anne
Published: August 2021

Using the expansion of the inverse of the Kostka matrix in terms of tabloids as presented by Eğecioğlu and Remmel, we show that the fusion coefficients can be expressed as an alternating sum over cylindric tableaux. Cylindric tableaux are skew tableaux with a certain cyclic symmetry. When the skew shape of the tableau has a cutting point, meaning that the cylindric skew shape is not connected, or if its weight has at most two parts, we give a positive combinatorial formula for the fusion coefficients. The proof uses a slight modification of a sign-reversing involution introduced by Remmel and Shimozono. We discuss how this approach may work in general.

En utilisant l'expansion de l'inverse de la matrice Kostka en termes de tabloïdes introduite par Eğecioğlu et Remmel, nous montrons que les coefficients de fusion peuvent être exprimés comme une somme alternée sur les tableaux cylindriques. Les tableaux cylindriques sont des tableaux qui présentent une certaine symétrie cyclique. Lorsque la forme du tableau a un point de coupure, ce qui signifie que la forme cylindrique n'est pas connecté, ou lorsque son poids a au plus deux parts, nous donnons une formule combinatoire positive des coefficients de fusion. La démonstration utilise une légère modification de l'involution qui change le signe introduite par Remmel et Shimozono. Nous discutons comment cette approche pourrait fonctionner en général.
Which Schubert varieties are local complete intersections?
Authors: Úlfarsson Henning , Woo Alexander
Published: August 2021

We characterize by pattern avoidance the Schubert varieties for $\mathrm{GL}_n$ which are local complete intersections (lci). For those Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighbourhoods at the identity. Although the statement of our characterization only requires ordinary pattern avoidance, showing that the Schubert varieties not satisfying our conditions are not lci appears to require working with more general notions of pattern avoidance. The Schubert varieties defined by inclusions, originally introduced by Gasharov and Reiner, turn out to be an important subclass, and we further develop some of their combinatorics. One application is a new formula for certain specializations of Schubert polynomials.

Nous caractérisons par l’évitement des motifs les variétés de Schubert qui sont localement des intersections complètes. Pour les variétés de Schubert qui sont localement des intersections complètes, nous donnons des ensembles explicites des polynômes qui définissent leurs voisinages à l’identité. Bien que notre caractérisation n'utilise que les motifs ordinaire, nous avons besoin des notions plus générales des motifs dans notre démonstration. Les variétés de Schubert définies par des inclusions, introduites par Gasharov et Reiner, sont une sous-classe importante, et nous développons davantage leurs combinatoire. Une application est une nouvelle formule pour une spécialisation des polynômes de Schubert.
Bijections for lattice paths between two boundaries
Authors: Elizalde Sergi , Rubey Martin
Published: August 2021

We prove that on the set of lattice paths with steps $N=(0,1)$ and $E=(1,0)$ that lie between two boundaries $B$ and $T$, the two statistics `number of $E$ steps shared with $B$' and `number of $E$ steps shared with $T$' have a symmetric joint distribution. We give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps $S=(0,-1)$ at prescribed $x$-coordinates. We also show that a similar equidistribution result for other path statistics follows from the fact that the Tutte polynomial of a matroid is independent of the order of its ground set. Finally, we extend the two theorems to $k$-tuples of paths between two boundaries, and we give some applications to Dyck paths, generalizing a result of Deutsch, and to pattern-avoiding permutations.

On montre que, sur l'ensemble des chemins avec des pas $N=(0,1)$ et $E=(1,0)$ qui se trouvent entre deux chemins donnés $B$ et $T$, les deux statistiques"`nombre des pas $E$ en commun avec $B$" et "nombre des pas $E$ en commun avec $T$" ont une distribution conjointe symétrique. On donne une involution qui échange ces deux statistiques, préserve quelques autres paramètres additionnels, et admet une généralisation à des chemins avec des pas $S=(0, -1)$ dans des positions données. On montre aussi un autre résultat d'équidistribution similaire, lié au polynôme de Tutte d'un matroïde. Finalement, on étend les deux théorèmes à $k$-tuples de chemins entre deux frontières, et on donne quelques applications aux chemins de Dyck, en généralisant un résultat de Deutsch, et aux permutations avec des motifs exclus.
Cartan invariant matrices for finite monoids
Authors: Thiéry Nicolas M.
Published: August 2021

Let $M$ be a finite monoid. In this paper we describe how the Cartan invariant matrix of the monoid algebra of $M$ over a field $\mathbb{K}$ of characteristic zero can be expressed using characters and some simple combinatorial statistic. In particular, it can be computed efficiently from the composition factors of the left and right class modules of $M$. When $M$ is aperiodic, this approach works in any characteristic, and generalizes to $\mathbb{K}$ a principal ideal domain like $\mathbb{Z}$. When $M$ is $\mathcal{R}$-trivial, we retrieve the formerly known purely combinatorial description of the Cartan matrix.

Soit $M$ un monoïde fini. Dans cet article, nous exprimons la matrice des invariants de Cartan de l'algèbre de $M$ sur un corps $\mathbb{K}$ de caractéristique zéro à l'aide de caractères et d'une statistique combinatoire simple. En particulier, elle peut être calculée efficacement à partir des facteurs de compositions des modules de classes à gauche et à droite de $M$. Lorsque $M$ est apériodique, cette approche se généralise à toute caractéristique et aux anneaux principaux comme $\mathbb{Z}$. Lorsque $M$ est $\mathcal{R}$-trivial, nous retrouvons la description combinatoire de la matrice de Cartan précédemment connue.
Well-spread sequences and edge-labellings with constant Hamilton-weight
Authors: Kayll Peter Mark
Published: August 2021

A sequence (a_i) of integers is \emphwell-spread if the sums a_i+a_j, for i
On an open problem of Green and Losonczy: exact enumeration of freely braided permutations
Authors: Mansour Toufik
Published: August 2021

Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par
Reconstruction Thresholds on Regular Trees
Authors: Martin James B.
Published: August 2021

We consider themodel of broadcasting on a tree, with binary state space, on theinfinite rooted tree $T^k$ in which each node has $k$ children. The root of the tree takesa random value $0$ or $1$, and then each node passes a value independently to each of its children according to a $2x2$ transition matrix $\mathbf{P}$. We say that reconstruction is possible if the values at the dth level of the tree contain non-vanishing information about the value at the root as $d→∞$. Extending a method of Brightwell and Winkler, we obtain new conditions under which reconstruction is impossible, both in the general case and in the special case $p_11=0$. The latter case is closely related to the hard-core model from statistical physics; a corollary of our results is that, for the hard-core model on the $(k+1)$-regular tree with activity $λ =1$, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any $k ≥ 2$.
Non-crossing trees revisited: cutting down and spanning subtrees
Authors: Panholzer Alois
Published: August 2021

Here we consider two parameters for random non-crossing trees: $\textit{(i)}$ the number of random cuts to destroy a size-$n$ non-crossing tree and $\textit{(ii)}$ the spanning subtree-size of $p$ randomly chosen nodes in a size-$n$ non-crossing tree. For both quantities, we are able to characterise for $n → ∞$ the limiting distributions. Non-crossing trees are almost conditioned Galton-Watson trees, and it has been already shown, that the contour and other usually associated discrete excursions converge, suitable normalised, to the Brownian excursion. We can interpret parameter $\textit{(ii)}$ as a functional of a conditioned random walk, and although we do not have such an interpretation for parameter $\textit{(i)}$, we obtain here limiting distributions, that are also arising as limits of some functionals of conditioned random walks.
Average properties of combinatorial problems and thermodynamics of spin models on graphs
Authors: Vezzani Alessandro , Cassi Davide , Burioni Raffaella
Published: August 2021

The study of thermodynamic properties of classical spin models on infinite graphs naturally leads to consider the new combinatorial problems of random-walks and percolation on the average. Indeed, spinmodels with O(n) continuous symmetry present spontaneous magnetization only on transient on the average graphs, while models with discrete symmetry (Ising and Potts) are spontaneously magnetized on graphs exhibiting percolation on the average. In this paper we define the combinatorial problems on the average, showing that they give rise to classifications of graph topology which are different from the ones obtained in usual (local) random-walks and percolation. Furthermore, we illustrate the theorem proving the correspondence between Potts model and average percolation.
The Saddle Point Method for the Integral of the Absolute Value of the Brownian Motion
Authors: Tolmatz Leonid
Published: August 2021

The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.
Some Algebraic Aspects of Morse Code Sequences
Authors: Cigler Johann
Published: August 2021

Morse code sequences are very useful to give combinatorial interpretations of various properties of Fibonacci numbers. In this note we study some algebraic and combinatorial aspects of Morse code sequences and obtain several q-analogues of Fibonacci numbers and Fibonacci polynomials and their generalizations.
Entropic repulsion on a rarefied wall
Authors: Fontes L. R. G. , Vachkovskaia M. , Yambartsev A.
Published: August 2021

We consider the motion of a discrete d-dimensional random surface interacting by exclusion with a rarefied wall. The dynamics is given by the serial harness process. We prove that the process delocalizes iff the mean number of visits to the set of sites where the wall is present by some random walk is infinite. In case where there is a delocalization, bounds on its speed are obtained.
Some results for monotonically labelled simply generated trees
Authors: Gittenberger Bernhard , Panholzer Alois
Published: August 2021

We consider simply generated trees, where the nodes are equipped with weakly monotone labellings with elements of $\{1, 2, \ldots, r\}$, for $r$ fixed. These tree families were introduced in Prodinger and Urbanek (1983) and studied further in Kirschenhofer (1984), Blieberger (1987), and Morris and Prodinger (2005). Here we give distributional results for several tree statistics (the depth of a random node, the ancestor-tree size and the Steiner-distance of $p$ randomly chosen nodes, the height of the $j$-st leaf, and the number of nodes with label $l$), which extend the existing results and also contain the corresponding results for unlabelled simply generated trees as the special case $r=1$.
Fountains, histograms, and q-identities
Authors: Paule Peter , Prodinger Helmut
Published: August 2021

We solve the recursion S_n=S_n-1-q^nS_n-p, both, explicitly, and in the limit for n→∞, proving in this way a formula due to Merlini and Sprugnoli. It is also discussed how computer algebra could be applied.
Near optimality of the discrete persistent access caching algorithm
Authors: Jelenković Predrag R. , Kang Xiaozhu , Radovanović Ana
Published: August 2021

Renewed interest in caching techniques stems from their application to improving the performance of the World Wide Web, where storing popular documents in proxy caches closer to end-users can significantly reduce the document download latency and overall network congestion. Rules used to update the collection of frequently accessed documents inside a cache are referred to as cache replacement algorithms. Due to many different factors that influence the Web performance, the most desirable attributes of a cache replacement scheme are low complexity and high adaptability to variability in Web access patterns. These properties are primarily the reason why most of the practical Web caching algorithms are based on the easily implemented Least-Recently-Used (LRU) cache replacement heuristic. In our recent paperJelenković and Radovanović (2004c), we introduce a new algorithm, termed Persistent Access Caching (PAC), that, in addition to desirable low complexity and adaptability, somewhat surprisingly achieves nearly optimal performance for the independent reference model and generalized Zipf's law request probabilities. Two drawbacks of the PAC algorithm are its dependence on the request arrival times and variable storage requirements. In this paper, we resolve these problems by introducing a discrete version of the PAC policy (DPAC) that, after a cache miss, places the requested document in the cache only if it is requested at least $k$ times among the last $m$, $m \geq k$, requests. However, from a mathematical perspective, due to the inherent coupling of the replacement decisions for different documents, the DPAC algorithm is considerably harder to analyze than the original PAC policy. In this regard, we develop a new analytical technique for estimating the performance of the DPAC rule. Using our analysis, we show that this algorithm is close to optimal even for small values of $k$ and $m$, and, therefore, adds negligible additional storage and processing complexity in comparison to the ordinary LRU policy.
Near―perfect non-crossing harmonic matchings in randomly labeled points on a circle
Authors: Balogh József , Pittel Boris , Salazar Gelasio
Published: August 2021

Consider a set $S$ of points in the plane in convex position, where each point has an integer label from $\{0,1,\ldots,n-1\}$. This naturally induces a labeling of the edges: each edge $(i,j)$ is assigned label $i+j$, modulo $n$. We propose the algorithms for finding large non―crossing $\textit{harmonic}$ matchings or paths, i. e. the matchings or paths in which no two edges have the same label. When the point labels are chosen uniformly at random, and independently of each other, our matching algorithm with high probability (w.h.p.) delivers a nearly―perfect matching, a matching of size $n/2 - O(n^{1/3}\ln n)$.
A repertoire for additive functionals of uniformly distributed m-ary search trees
Authors: fill james Allen , Kapur Nevin
Published: August 2021

Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on $m$-ary search trees on $n$ keys with toll sequence $(i) n^α$ with $α ≥ 0 (α =0$ and $α =1$ correspond roughly to the space requirement and total path length, respectively); $(ii) ln \binom{n} {m-1}$, which corresponds to the so-called shape functional; and $(iii) $$1$$_{n=m-1}$, which corresponds to the number of leaves.
Rapidly mixing chain and perfect sampler for logarithmic separable concave distributions on simplex
Authors: Kijima Shuji , Matsui Tomomi
Published: August 2021

In this paper, we are concerned with random sampling of an n dimensional integral point on an $(n-1)$ dimensional simplex according to a multivariate discrete distribution. We employ sampling via Markov chain and propose two "hit-and-run'' chains, one is for approximate sampling and the other is for perfect sampling. We introduce an idea of alternating inequalities  and show that a logarithmic separable concave function satisfies the alternating inequalities. If a probability function satisfies alternating inequalities, then our chain for approximate sampling mixes in $\textit{O}(n^2 \textit{ln}(Kɛ^{-1}))$, namely $(1/2)n(n-1) \textit{ln}(K ɛ^{-1})$, where $K$ is the side length of the simplex and $ɛ (0<ɛ<1)$ is an error rate. On the same condition, we design another chain and a perfect sampler based on monotone CFTP (Coupling from the Past). We discuss a condition that the expected number of total transitions of the chain in the perfect sampler is bounded by $\textit{O}(n^3 \textit{ln}(Kn))$.
Asymptotics of Riordan arrays
Authors: Wilson Mark C.
Published: August 2021

The machinery of Riordan arrays has been used recently by several authors. We show how meromorphic singularity analysis can be used to provide uniform bivariate asymptotic expansions, in the central regime, for a generalization of these arrays. We show how to do this systematically, for various descriptions of the array. Several examples from recent literature are given.
An application of results by Hardy, Ramanujan and Karamata to Ackermannian functions
Authors: Weiermann Andreas
Published: August 2021

The Ackermann function is a fascinating and well studied paradigm for a function which eventually dominates all primitive recursive functions. By a classical result from the theory of recursive functions it is known that the Ackermann function can be defined by an unnested or descent recursion along the segment of ordinals below ω ^ω (or equivalently along the order type of the polynomials under eventual domination). In this article we give a fine structure analysis of such a Ackermann type descent recursion in the case that the ordinals below ω ^ω are represented via a Hardy Ramanujan style coding. This paper combines number-theoretic results by Hardy and Ramanujan, Karamata's celebrated Tauberian theorem and techniques from the theory of computability in a perhaps surprising way.
Crooked Maps in Finite Fields
Authors: Kyureghyan Gohar
Published: August 2021

We consider the maps $f:\mathbb{F}_{2^n} →\mathbb{F}_{2^n}$ with the property that the set $\{ f(x+a)+ f(x): x ∈F_{2^n}\}$ is a hyperplane or a complement of hyperplane for every $a ∈\mathbb{F}_{2^n}^*$. The main goal of the talk is to show that almost all maps $f(x) = Σ_{b ∈B}c_b(x+b)^d$, where $B ⊂\mathbb{F}_{2^n}$ and $Σ_{b ∈B}c_b ≠0$, are not of that type. In particular, the only such power maps have exponents $2^i+2^j$ with $gcd(n, i-j)=1$. We give also a geometrical characterization of this maps.
A characterization of extremal graphs with no matching-cut
Authors: Bonsma Paul
Published: August 2021

A graph is called (matching-)immune if it has no edge cut that is also a matching. Farley and Proskurowski proved that for all immune graphs $G=(V,E)$, $|E|≥\lceil 3(|V|-1)/2\rceil$ , and constructed a large class of immune graphs that attain this lower bound for every value of $|V(G)|$, called $ABC$ graphs. They conjectured that every immune graph that attains this lower bound is an $ABC$ graph. We present a proof of this conjecture.
Supermodularity on chains and complexity of maximum constraint satisfaction
Authors: Deineko Vladimir , Jonsson Peter , Klasson Mikael , Krokhin Andrei
Published: August 2021

In the maximum constraint satisfaction problem ($\mathrm{Max \; CSP}$), one is given a finite collection of (possibly weighted) constraints on overlapping sets of variables, and the goal is to assign values from a given finite domain to the variables so as to maximise the number (or the total weight) of satisfied constraints. This problem is $\mathrm{NP}$-hard in general so it is natural to study how restricting the allowed types of constraints affects the complexity of the problem. In this paper, we show that any $\mathrm{Max \; CSP}$ problem with a finite set of allowed constraint types, which includes all constants (i.e. constraints of the form $x=a$), is either solvable in polynomial time or is $\mathrm{NP}$-complete. Moreover, we present a simple description of all polynomial-time solvable cases of our problem. This description uses the well-known combinatorial property of supermodularity.
Permutations with short monotone subsequences
Authors: Romik Dan
Published: August 2021

We consider permutations of $1,2,...,n^2$ whose longest monotone subsequence is of length $n$ and are therefore extremal for the Erdős-Szekeres Theorem. Such permutations correspond via the Robinson-Schensted correspondence to pairs of square $n \times n$ Young tableaux. We show that all the bumping sequences are constant and therefore these permutations have a simple description in terms of the pair of square tableaux. We deduce a limit shape result for the plot of values of the typical such permutation, which in particular implies that the first value taken by such a permutation is with high probability $(1+o(1))n^2/2$.
Hypertree-Width and Related Hypergraph Invariants
Authors: Adler Isolde , Gottlob Georg , Grohe Martin
Published: August 2021

We study the notion of hypertree-width of hypergraphs. We prove that, up to a constant factor, hypertree-width is the same as a number of other hypergraph invariants that resemble graph invariants such as bramble-number, branch-width, linkedness, and the minimum number of cops required to win Seymour and Thomas's robber and cops game.
The game of arboricity
Authors: Bartnicki Tomasz , Grytczuk Jaroslaw , Kierstead Hal
Published: August 2021

Using a fixed set of colors $C$, Ann and Ben color the edges of a graph $G$ so that no monochromatic cycle may appear. Ann wins if all edges of $G$ have been colored, while Ben wins if completing a coloring is not possible. The minimum size of $C$ for which Ann has a winning strategy is called the $\textit{game arboricity}$ of $G$, denoted by $A_g(G)$. We prove that $A_g(G) \leq 3k$ for any graph $G$ of arboricity $k$, and that there are graphs such that $A_g(G) \geq 2k-2$. The upper bound is achieved by a suitable version of the activation strategy, used earlier for the vertex coloring game. We also provide other strategie based on induction.
Removing Even Crossings
Authors: Pelsmajer Michael J. , Schaefer Marcus , Štefankovič Daniel
Published: August 2021

An edge in a drawing of a graph is called $\textit{even}$ if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (not using Kuratowski's theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most $3$. We begin with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.
On the $L(p,1)$-labelling of graphs
Authors: Gonçalves Daniel
Published: August 2021

In this paper we improve the best known bound for the $L(p,1)$-labelling of graphs with given maximal degree.
Representations of Edge Intersection Graphs of Paths in a Tree
Authors: Golumbic Martin Charles , Lipshteyn Marina , Stern Michal
Published: August 2021

Let $\mathcal{P}$ be a collection of nontrivial simple paths in a tree $T$. The edge intersection graph of $\mathcal{P}$, denoted by EPT($\mathcal{P}$), has vertex set that corresponds to the members of $\mathcal{P}$, and two vertices are joined by an edge if the corresponding members of $\mathcal{P}$ share a common edge in $T$. An undirected graph $G$ is called an edge intersection graph of paths in a tree, if $G = EPT(\mathcal{P})$ for some $\mathcal{P}$ and $T$. The EPT graphs are useful in network applications. Scheduling undirected calls in a tree or assigning wavelengths to virtual connections in an optical tree network are equivalent to coloring its EPT graph. It is known that recognition and coloring of EPT graphs are NP-complete problems. However, the EPT graphs restricted to host trees of vertex degree 3 are precisely the chordal EPT graphs, and therefore can be colored in polynomial time complexity. We prove a new analogous result that weakly chordal EPT graphs are precisely the EPT graphs with host tree restricted to degree 4. This also implies that the coloring of the edge intersection graph of paths in a degree 4 tree is polynomial. We raise a number of intriguing conjectures regarding related families of graphs.
Colouring random geometric graphs
Authors: McDiarmid Colin J. H. , Müller Tobias
Published: August 2021

A random geometric graph $G_n$ is obtained as follows. We take $X_1, X_2, \ldots, X_n ∈\mathbb{R}^d$ at random (i.i.d. according to some probability distribution ν on $\mathbb{R}^d$). For $i ≠j$ we join $X_i$ and $X_j$ by an edge if $║X_i - X_j ║< r(n)$. We study the properties of the chromatic number $χ _n$ and clique number $ω _n$ of this graph as n becomes large, where we assume that $r(n) →0$. We allow any choice $ν$ that has a bounded density function and $║. ║$ may be any norm on $ℝ^d$. Depending on the choice of $r(n)$, qualitatively different types of behaviour can be observed. We distinguish three main cases, in terms of the key quantity $n r^d$ (which is a measure of the average degree). If $r(n)$ is such that $\frac{nr^d}{ln n} →0$ as $n →∞$ then $\frac{χ _n}{ ω _n} →1$ almost surely. If n $\frac{r^d }{\ln n} →∞$ then $\frac{χ _n }{ ω _n} →1 / δ$ almost surely, where $δ$ is the (translational) packing density of the unit ball $B := \{ x ∈ℝ^d: ║x║< 1 \}$ (i.e. $δ$ is the proportion of $d$-space that can be filled with disjoint translates of $B$). If $\frac{n r^d }{\ln n} →t ∈(0,∞)$ then $\frac{χ _n }{ ω _n}$ tends almost surely to a constant that can be bounded in terms of $δ$ and $t$. These results extend earlier work of McDiarmid and Penrose. The proofs in fact yield separate expressions for $χ _n$ and $ω _n$. We are also able to prove a conjecture by Penrose. This states that when $\frac{n r^d }{ln n} →0$ then the clique number becomes focussed on two adjacent integers, meaning that there exists a sequence $k(n)$ such that $\mathbb{P}( ω _n ∈\{k(n), k(n)+1\}) →1$ as $n →∞$. The analogous result holds for the chromatic number (and for the maximum degree, as was already shown by Penrose in the uniform case).
Spanning paths in hypercubes
Authors: Dvořák Tomáš , Gregor Petr , Koubek Václav
Published: August 2021

Given a family $\{u_i,v_i\}_{i=1}^k$ of pairwise distinct vertices of the $n$-dimensional hypercube $Q_n$ such that the distance of $u_i$ and $v_i$ is odd and $k \leq n-1$, there exists a family $\{P_i\}_{i=1}^k$ of paths such that $u_i$ and $v_i$ are the endvertices of $P_i$ and $\{V(P_i)\}_{i=1}^k$ partitions $V(Q_n)$. This holds for any $n \geq 2$ with one exception in the case when $n=k+1=4$. On the other hand, for any $n \geq 3$ there exist $n$ pairs of vertices satisfying the above condition for which such a family of spanning paths does not exist. We suggest further generalization of this result and explore a relationship to the problem of hamiltonicity of hypercubes with faulty vertices.
On the Minimum Number of Completely 3-Scrambling Permutations
Authors: Tarui Jun
Published: August 2021

A family $\mathcal{P} = \{\pi_1, \ldots , \pi_q\}$ of permutations of $[n]=\{1,\ldots,n\}$ is $\textit{completely}$ $k$-$\textit{scrambling}$ [Spencer, 1972; Füredi, 1996] if for any distinct $k$ points $x_1,\ldots,x_k \in [n]$, permutations $\pi_i$'s in $\mathcal{P}$ produce all $k!$ possible orders on $\pi_i (x_1),\ldots, \pi_i(x_k)$. Let $N^{\ast}(n,k)$ be the minimum size of such a family. This paper focuses on the case $k=3$. By a simple explicit construction, we show the following upper bound, which we express together with the lower bound due to Füredi for comparison. $\frac{2}{ \log _2e} \log_2 n \leq N^{\ast}(n,3) \leq 2\log_2n + (1+o(1)) \log_2 \log _2n$. We also prove the existence of $\lim_{n \to \infty} N^{\ast}(n,3) / \log_2 n = c_3$. Determining the value $c_3$ and proving the existence of $\lim_{n \to \infty} N^{\ast}(n,k) / \log_2 n = c_k$ for $k \geq 4$ remain open.
Hamiltonian Chains in Hypergraphs
Authors: Katona Gyula Y.
Published: August 2021

Hamiltionian chain is a generalisation of hamiltonian cycles for hypergraphs. Among the several possible ways of generalisations this is probably the most strong one, it requires the strongest structure. Since there are many interesting questions about hamiltonian cycles in graphs, we can try to answer these questions for hypergraphs, too. In the present article we give a survey on results about such questions.
Matchings and Hamilton cycles in hypergraphs
Authors: Kühn Daniela , Osthus Deryk
Published: August 2021

It is well known that every bipartite graph with vertex classes of size $n$ whose minimum degree is at least $n/2$ contains a perfect matching. We prove an analogue of this result for uniform hypergraphs. We also provide an analogue of Dirac's theorem on Hamilton cycles for $3$-uniform hypergraphs: We say that a $3$-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. We prove that for every $\varepsilon > 0$ there is an $n_0$ such that every $3$-uniform hypergraph of order $n \geq n_0$ whose minimum degree is at least $n/4+ \varepsilon n$ contains a Hamilton cycle. Our bounds on the minimum degree are essentially best possible.
A Probabilistic Counting Lemma for Complete Graphs
Authors: Gerke Stefanie , Marciniszyn Martin , Steger Angelika
Published: August 2021

We prove the existence of many complete graphs in almost all sufficiently dense partitions obtained by an application of Szemerédi's Regularity Lemma. More precisely, we consider the number of complete graphs $K_{\ell}$ on $\ell$ vertices in $\ell$-partite graphs where each partition class consists of $n$ vertices and there is an $\varepsilon$-regular graph on $m$ edges between any two partition classes. We show that for all $\beta > $0, at most a $\beta^m$-fraction of graphs in this family contain less than the expected number of copies of $K_{\ell}$ provided $\varepsilon$ is sufficiently small and $m \geq Cn^{2-1/(\ell-1)}$ for a constant $C > 0$ and $n$ sufficiently large. This result is a counting version of a restricted version of a conjecture by Kohayakawa, Łuczak and Rödl and has several implications for random graphs.
Kernel perfect and critical kernel imperfect digraphs structure
Authors: Galeana-Sánchez Hortensia , Guevara Mucuy-Kak
Published: August 2021

A kernel $N$ of a digraph $D$ is an independent set of vertices of $D$ such that for every $w \in V(D)-N$ there exists an arc from $w$ to $N$. If every induced subdigraph of $D$ has a kernel, $D$ is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If $F$ is a set of arcs of $D$, a semikernel modulo $F$, $S$ of $D$ is an independent set of vertices of $D$ such that for every $z \in V(D)- S$ for which there exists an $Sz-$arc of $D-F$, there also exists an $zS-$arc in $D$. In this talk some structural results concerning critical kernel imperfect and sufficient conditions for a digraph to be a critical kernel imperfect digraph are presented.
Tilings from some non-irreducible, Pisot substitutions
Authors: Ito Shunji , Ei Hiromi
Published: August 2021

A generating method of self-affine tilings for Pisot, unimodular, irreducible substitutions, as well as the fact that the associated substitution dynamical systems are isomorphic to rotations on the torus are established in P. Arnoux and S. Ito. The aim of this paper is to extend these facts in the case where the characteristic polynomial of a substitution is non-irreducible for a special class of substitutions on five letters. Finally we show that the substitution dynamical systems for this class are isomorphic to induced transformations of rotations on the torus.
Undecidable problems concerning densities of languages
Authors: Kozik Jakub
Published: August 2021

In this paper we prove that the question whether a language presented by a context free grammar has density, is undecidable. Moreover we show that there is no algorithm which, given two unambiguous context free grammars on input, decides whether the language defined by the first grammar has a relative density in the language defined by the second one. Our techniques can be extended to show that this problem is undecidable even for languages given by grammars from $LL(k)$ (for sufficiently large fixed $k ∈ \mathbb{N} )$.
Solving equations over small unary algebras
Authors: Broniek Przemyslaw
Published: August 2021

We consider the problem of solving a system of polynomial equations over fixed algebra $A$ which we call MPolSat($A$). We restrict ourselves to unary algebras and give a partial characterization of complexity of MPolSat($A$). We isolate a preorder $P(A)$ to show that when $A$ has at most 3 elements then MPolSat($A$) is in $P$ when width of $P(A)$ is at most 2 and is NP-complete otherwise. We show also that if $P ≠ NP$ then the class of unary algebras solvable in polynomial time is not closed under homomorphic images.
Polyominoes determined by permutations
Authors: Fanti I. , Frosini A. , Grazzini E. , Pinzani R. , Rinaldi S.
Published: August 2021

In this paper we consider the class of $\textit{permutominoes}$, i.e. a special class of polyominoes which are determined by a pair of permutations having the same size. We give a characterization of the permutations associated with convex permutominoes, and then we enumerate various classes of convex permutominoes, including parallelogram, directed-convex, and stack ones.
Around the root of random multidimensional quadtrees
Authors: Labelle Gilbert , Laforest Louise , Provençal Xavier
Published: August 2021

We analyse the distribution of the root pattern of randomly grown multidimensional point quadtrees. In particular, exact, recursive and asymptotic formulas are given for the expected arity of the root.
Samples of geometric random variables with multiplicity constraints
Authors: Archibald Margaret , Knopfmacher Arnold
Published: August 2021

We investigate the probability that a sample $\Gamma=(\Gamma_1,\Gamma_2,\ldots,\Gamma_n)$ of independent, identically distributed random variables with a geometric distribution has no elements occurring exactly $j$ times, where $j$ belongs to a specified finite $\textit{'forbidden set'}$ $A$ of multiplicities. Specific choices of the set $A$ enable one to determine the asymptotic probabilities that such a sample has no variable occuring with multiplicity $b$, or which has all multiplicities greater than $b$, for any fixed integer $b \geq 1$.
Trees with product-form random weights
Authors: Borovkov Konstantin , Vatutin Vladimir
Published: August 2021

We consider growing random recursive trees in random environment, in which at each step a new vertex is attached according to a probability distribution that assigns the tree vertices masses proportional to their random weights.The main aim of the paper is to study the asymptotic behavior of the mean numbers of outgoing vertices as the number of steps tends to infinity, under the assumption that the random weights have a product form with independent identically distributed factors.
Left and right length of paths in binary trees or on a question of Knuth
Authors: Panholzer Alois
Published: August 2021

We consider extended binary trees and study the common right and left depth of leaf $j$, where the leaves are labelled from left to right by $0, 1, \ldots, n$, and the common right and left external pathlength of binary trees of size $n$. Under the random tree model, i.e., the Catalan model, we characterize the common limiting distribution of the suitably scaled left depth and the difference between the right and the left depth of leaf $j$ in a random size-$n$ binary tree when $j \sim \rho n$ with $0< \rho < 1$, as well as the common limiting distribution of the suitably scaled left external pathlength and the difference between the right and the left external pathlength of a random size-$n$ binary tree.
Random assignment and shortest path problems
Authors: Wästlund Johan
Published: August 2021

We explore a similarity between the $n$ by $n$ random assignment problem and the random shortest path problem on the complete graph on $n+1$ vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We give direct proofs of the analogs for the shortest path problem of some results established by D. Aldous in connection with his $\zeta (2)$ limit theorem for the assignment problem.
An invariance principle for random planar maps
Authors: Miermont Grégory
Published: August 2021

We show a new invariance principle for the radius and other functionals of a class of conditioned `Boltzmann-Gibbs'-distributed random planar maps. It improves over the more restrictive case of bipartite maps that was discussed in Marckert and Miermont (2006). As in the latter paper, we make use of a bijection between planar maps and a class of labelled multitype trees, due to Bouttier et al. (2004). We also rely on an invariance principle for multitype spatial Galton-Watson trees, which is proved in a companion paper.
Expected values of statistics on permutation tableaux
Authors: Corteel Sylvie , Hitczenko Pawel
Published: August 2021

Permutation tableaux are new objects that were introduced by Postnikov in the context of enumeration of the totally positive Grassmannian cells. They are known to be in bijection with permutations and recently, they have been connected to PASEP model used in statistical physics. Properties of permutation tableaux became a focus of a considerable research activity. In this paper we study properties of basic statistics defined on permutation tableaux. We present a simple and unified approach based on probabilistic techniques and use it to compute the expected values of basic statistics defined on permutation tableaux. We also provide a non―bijective and very simple proof that there are n! permutation tableaux of length n.
Limit laws for a class of diminishing urn models.
Authors: Kuba Markus , Panholzer Alois
Published: August 2021

In this work we analyze a class of diminishing 2×2 Pólya-Eggenberger urn models with ball replacement matrix M given by $M= \binom{ -a \,0}{c -d}, a,d∈\mathbb{N}$ and $c∈\mathbb{N} _0$. We obtain limit laws for this class of 2×2 urns by giving estimates for the moments of the considered random variables. As a special instance we obtain limit laws for the pills problem, proposed by Knuth and McCarthy, which corresponds to the special case $a=c=d=1$. Furthermore, we also obtain limit laws for the well known sampling without replacement urn, $a=d=1$ and $c=0$, and corresponding generalizations, $a,d∈\mathbb{N}$ and $c=0$.
Asynchronous Cellular Automata and Brownian Motion
Authors: Chassaing Philippe , Gerin Lucas
Published: August 2021

This paper deals with some very simple interacting particle systems, \emphelementary cellular automata, in the fully asynchronous dynamics: at each time step, a cell is randomly picked, and updated. When the initial configuration is simple, we describe the asymptotic behavior of the random walks performed by the borders of the black/white regions. Following a classification introduced by Fatès \emphet al., we show that four kinds of asymptotic behavior arise, two of them being related to Brownian motion.
Distributional asymptotics in the analysis of algorithms: Periodicities and discretization
Authors: Grübel Rudolf
Published: August 2021

It is well known that many distributions that arise in the analysis of algorithms have an asymptotically fluctuating behaviour in the sense that we do not have 'full' convergence, but only convergence along suitable subsequences as the size of the input to the algorithm tends to infinity. We are interested in constructions that display such behaviour via an ordinarily convergent background process in the sense that the periodicities arise from this process by deterministic transformations, typically involving discretization as a decisive step. This leads to structural representations of the resulting family of limit distributions along subsequences, which in turn may give access to their properties, such as the tail behaviour (unsuccessful search in digital search trees) or the dependence on parameters of the algorithm (success probability in a selection algorithm).
The height of watermelons with wall - Extended Abstract
Authors: Feierl Thomas
Published: August 2021

We derive asymptotics for the moments of the height distribution of watermelons with $p$ branches with wall. This generalises a famous result by de Bruijn, Knuth and Rice on the average height of planted plane trees, and a result by Fulmek on the average height of watermelons with two branches.
On the Ehrenfeucht-Mycielski Balance Conjecture
Authors: Kieffer John C. , Szpankowski W.
Published: August 2021

In 1992, A. Ehrenfeucht and J. Mycielski defined a seemingly pseudorandom binary sequence which has since been termed the EM-sequence. The balance conjecture for the EM-sequence, still open, is the conjecture that the sequence of EM-sequence initial segment averages converges to $1/2$. In this paper, we do not prove the balance conjecture but we do make some progress concerning it, namely, we prove that every limit point of the aforementioned sequence of averages lies in the interval $[1/4,3/4]$, improving the best previous result that every such limit point belongs to the interval $[0.11,0.89]$. Our approach is novel and exploits an analysis of the growth behavior as $n \to \infty$ of the rooted tree formed by the binary strings appearing at least twice as substrings of the length $n$ initial segment of the EM-sequence.
Message passing for the coloring problem: Gallager meets Alon and Kahale
Authors: Ben-Shimon Sonny , Vilenchik Dan
Published: August 2021

Message passing algorithms are popular in many combinatorial optimization problems. For example, experimental results show that \emphsurvey propagation (a certain message passing algorithm) is effective in finding proper k-colorings of random graphs in the near-threshold regime. In 1962 Gallager introduced the concept of Low Density Parity Check (LDPC) codes, and suggested a simple decoding algorithm based on message passing. In 1994 Alon and Kahale exhibited a coloring algorithm and proved its usefulness for finding a k-coloring of graphs drawn from a certain planted-solution distribution over k-colorable graphs. In this work we show an interpretation of Alon and Kahale's coloring algorithm in light of Gallager's decoding algorithm, thus showing a connection between the two problems - coloring and decoding. This also provides a rigorous evidence for the usefulness of the message passing paradigm for the graph coloring problem.
Average Redundancy for Known Sources: Ubiquitous Trees in Source Coding
Authors: Szpankowski Wojciech
Published: August 2021

Analytic information theory aims at studying problems of information theory using analytic techniques of computer science and combinatorics. Following Hadamard's precept, these problems are tackled by complex analysis methods such as generating functions, Mellin transform, Fourier series, saddle point method, analytic poissonization and depoissonization, and singularity analysis. This approach lies at the crossroad of computer science and information theory. In this survey we concentrate on one facet of information theory (i.e., source coding better known as data compression), namely the $\textit{redundancy rate}$ problem. The redundancy rate problem determines by how much the actual code length exceeds the optimal code length. We further restrict our interest to the $\textit{average}$ redundancy for $\textit{known}$ sources, that is, when statistics of information sources are known. We present precise analyses of three types of lossless data compression schemes, namely fixed-to-variable (FV) length codes, variable-to-fixed (VF) length codes, and variable-to-variable (VV) length codes. In particular, we investigate average redundancy of Huffman, Tunstall, and Khodak codes. These codes have succinct representations as $\textit{trees}$, either as coding or parsing trees, and we analyze here some of their parameters (e.g., the average path from the root to a leaf).
Distribution of the Number of Encryptions in Revocation Schemes for Stateless Receivers
Authors: Eagle Christopher , Gao Zhicheng , Omar Mohamed , Panario Daniel , Richmond Bruce
Published: August 2021

We study the number of encryptions necessary to revoke a set of users in the complete subtree scheme (CST) and the subset-difference scheme (SD). These are well-known tree based broadcast encryption schemes. Park and Blake in: Journal of Discrete Algorithms, vol. 4, 2006, pp. 215―238, give the mean number of encryptions for these schemes. We continue their analysis and show that the limiting distribution of the number of encryptions for these schemes is normal. This implies that the mean numbers of Park and Blake are good estimates for the number of necessary encryptions used by these schemes.