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Cooperation among kin is common in animal societies. Kin groups may form by individuals directly discriminating relatives based on kin recognition cues, or form passively through natal philopatry and limited dispersal. We describe the genetic landscape for a primitively eusocial wasp, Polistes dominula, and ask whether individuals choose cooperative partners that are nearby and/or that are genetic relatives. Firstly, we genotyped an entire sub-population of 1361 wasps and found genetic structuring on an extremely fine scale: the probability of finding genetic relatives decreases exponentially within just a few meters of an individual’s nest. At the same time, however, we found a lack of genetic structuring between natural nest aggregations within the population. Secondly, in a separate dataset where ~2000 wasps were genotyped, we show that wasps forced experimentally to make a new nest choice tended to choose new nests near to their original nests, and that these nests tended to contain some full sisters. However, a significant fraction of wasps chose nests that did not contain sisters, despite sisters being present in nearby nests. Although we cannot rule out a role for direct kin recognition or natal nest-mate recognition, our data suggest that kin groups may form via a philopatric rule-of-thumb, whereby wasps simply select groups and nesting sites that are nearby. The result is that most subordinate helpers obtain indirect fitness benefits by breeding cooperatively. [ABSTRACT FROM AUTHOR]

Abstract: We show that paper-folding words contain arbitrarily large abelian powers. [Copyright &y& Elsevier]

This paper formalizes the stepping stone problem introduced Ladoucer and Rebenstock [G] to the setting of simple graphs. This paper considers the set of functions from the vertices of our graph to N, which assign a fixed number of 1's to some vertices and assign higher numbers to other vertices by adding up their neighbors' assignments. The stepping stone solution is defined as an element obtained from the argmax of the maxima of these functions, and the maxima as its growth. This work is organized into work on the bounded and unbounded degree graph cases. In the bounded case, sufficient conditions are obtained for superlinear and sublinear stepping stone solution growth. Furthermore this paper demonstrates the existence of a basis of graphs which characterizes superlinear growth. In the unbounded case, properly sublinear and superlinear stepping stone solution growth are obtained. [ABSTRACT FROM AUTHOR]

The method of hypergraph containers, which was introduced several years ago by Balogh, Morris, and Samotij, and independently by Saxton and Thomason, has proved to be an extremely useful tool in the study of various monotone graph properties. In particular, a fairly straightforward application of this technique allows one to locate, for each non-bipartite graph H, the threshold at which the distribution of edges in a typical H-free graph with a given number of edges undergoes a transition from 'random-like' to 'structured'. On the other hand, for nonmonotone hereditary graph properties, the standard version of this method does not allow one to establish even the existence of such a threshold. In this paper we introduce a refinement of the container method that takes into account the asymmetry between edges and non-edges in a sparse member of a hereditary graph property. As an application, we determine the approximate structure of a typical graph with n vertices, m edges, and no induced copy of the 4-cycle, for each function m D m.n/ satisfying n4=3.log n/4 6 m n2. We show that almost all such graphs G have the following property: the vertex set of G can be partitioned into an 'almost independent' set (a set with o.m/ edges) and an 'almost-clique' (a set inducing a subgraph with density 1-a The lower bound on m is optimal up to a polylogarithmic factor, as standard arguments show that if n then almost all such graphs are 'random-like'. As a further consequence, we deduce that the random graph G.n; p/ conditioned to contain no induced 4-cycles undergoes phase transitions at p=n -2/3+o(1) p=n -2/3+o(1). [ABSTRACT FROM AUTHOR]

This paper proposes explicit solutions for the algebraic Riccati matrix equation. For single‐input systems in controllable canonical form, the explicit Hermitian solutions of the non‐homogeneous Riccati equation are obtained using the entries of the system matrix, the closed‐loop system matrix, and the weighting matrix. The unknown entries of the closed‐loop system matrix are solved by scalar quadratic equations. For a homogeneous Riccati equation with a zero weighting matrix, the explicit solutions are proposed analytically in terms of the system eigenvalues. The advantages of the explicit solutions are threefold: first, if the system is controllable, the solution is directly given and the invariant subspaces of the Hamiltonian matrix are not required; second, if the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms; third, for a real system in the controllable canonical form, the non‐negativity can be analysed for the explicit almost stabilizing solution. [ABSTRACT FROM AUTHOR]

A matching of a graph is a set of its edges that pairwise do not have common vertices. An important parameter of graphs, which is used in mathematical chemistry, is the Hosoya index, defined as the number of their matchings. Previously, the problems of maximizing this index were considered and completely solved for -vertex trees with two, three and four leaves for any sufficiently large . In the present paper, a similar problem is completely solved for 5-leaved trees with and for 6-leaved trees with . [ABSTRACT FROM AUTHOR]

A set C of vertices in a graph G = (V, E) is an identifying code if it is dominating and any two vertices of V are dominated by distinct sets of codewords. This paper presents a survey of Iiro Honkala's contributions to the study of identifying codes with respect to several aspects: complexity of computing an identifying code, combinatorics in binary Hamming spaces, infinite grids, relationships between identifying codes and usual parameters in graphs, structural properties of graphs admitting identifying codes, and number of optimal identifying codes. [ABSTRACT FROM AUTHOR]

Operations in the theory of graphs has a substantial inuence in the analytical and factual dimensions of the domain. In the realm of chemical graph theory, topological descriptor serves as a comprehensive graph invariant linked with a specific molecular structure. The study on the Sombor index is initiated recently by Ivan Gutman. The triangle parallel graph comprises of the edges of subdivision graph along with the edges of the original graph. In this paper, we make use of combinatorial inequalities related with the vertices, edges and the neighborhood concepts as well as the other topological descriptors in the computations for the determination of bounds of Sombor index for certain corona products involving the triangle parallel graph. [ABSTRACT FROM AUTHOR]

Catalan sequence initially introduced by Euler but named after Eugene Catalan has found abundant applications in the fields of probability theory, number theory, analysis, combinatorics etc. Recently ˙Ilkhan formulated the matrix associated with Catalan numbers and studied its domain in c0(C~) and c(C~). In this paper, by employing Catalan matrix defined by Catalan sequence and convergence via ideal, we establish Catalan sequence spaces c I 0 (C~), c I (C~) and ℓ I ∞(C~), study the properties of these corresponding spaces and establish inclusion relations. [ABSTRACT FROM AUTHOR]

A D-decomposition of a graph (or digraph) G is a partition of the edge set (or arc set) of G into subsets, where each subset induces a copy of the fixed graph D. Graph decomposition finds motivation in numerous practical applications, particularly in the realm of symmetric graphs, where these decompositions illuminate intricate symmetrical patterns within the graph, aiding in various fields such as network design, and combinatorial mathematics, among various others. Of particular interest is the case where G is K * λ K v * , the λ -fold complete symmetric digraph on v vertices, that is, the digraph with λ directed edges in each direction between each pair of vertices. For a given digraph D, the set of all values v for which K * λ K v * has a D-decomposition is called the λ -fold spectrum of D. An eight-cycle has 22 non-isomorphic orientations. The λ -fold spectrum problem has been solved for one of these oriented cycles. In this paper, we provide a complete solution to the λ -fold spectrum problem for each of the remaining 21 orientations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of normalized distance Laplacian matrices for signed graphs corresponding to the two signed distances defined for signed graphs. We characterize balance in signed graphs using these matrices and compare the normalized distance Laplacian spectral radius of signed graphs with that of all-negative signed graphs. Also we characterize the signed graphs having maximum normalized distance Laplacian spectral radius. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some of the key properties of sum-free subsets of abelian groups. Our discussion has been designed with a broader readership in mind and is hence not overly technical. We consider answers to questions like the following: How many sum-free subsets are there in a given abelian group G? What are its sum-free subsets of maximum cardinality? What is the maximum cardinality of these sum-free subsets? What does a typical sum-free subset of G look like? [ABSTRACT FROM AUTHOR]

Developing stability analysis methods for modern dynamical system solutions has been a significant challenge in the field. This study aims to formulate a qualitative analysis approach for the monotone stability region of a specific solution to a single differential equation within a dynamical system. The system in question comprises two first-order nonlinear ordinary differential equations of a particular kind. The method proposed hinges on applying elements of combinatorics to the traditional mathematical investigation of a function with a single independent variable. This approach enables the exact determination of the different qualitative scenarios in which the desired solution changes, under the assumption that the function values monotonically diminish from a specified value down to a finite zero. This paper outlines the creation and decomposition of the monotone stability region associated with the solution under consideration. [ABSTRACT FROM AUTHOR]

Mathematical modeling of the emergent dynamics of gene regulatory networks (GRN) faces a double challenge of (a) dependence of model dynamics on parameters, and (b) lack of reliable experimentally determined parameters. In this paper we compare two complementary approaches for describing GRN dynamics across unknown parameters: (1) parameter sampling and resulting ensemble statistics used by RACIPE (RAndom CIrcuit PErturbation), and (2) use of rigorous analysis of combinatorial approximation of the ODE models by DSGRN (Dynamic Signatures Generated by Regulatory Networks). We find a very good agreement between RACIPE simulation and DSGRN predictions for four different 2- and 3-node networks typically observed in cellular decision making. This observation is remarkable since the DSGRN approach assumes that the Hill coefficients of the models are very high while RACIPE assumes the values in the range 1-6. Thus DSGRN parameter domains, explicitly defined by inequalities between systems parameters, are highly predictive of ODE model dynamics within a biologically reasonable range of parameters. [ABSTRACT FROM AUTHOR]

The cutwidth minimization problem consists of finding an arrangement of the vertices of a graph G on a line Pn with n = |V (G)| vertices, in such a way that the maximum number of edges between each pair of consecutive vertices is minimized. A graph G with cutwidth k (k ≥ 1) is edge-maximal if c(G + uv) > k for any uv ∈ {uv: u, v ∈ V (G) and uv ∉ E(G)}. In this paper, we provide a complete insight to structural properties of edgemaximal graphs with cutwidth at most 3. [ABSTRACT FROM AUTHOR]

MALTA is part of the Depleted Monolithic Active Pixel sensors designed in Tower 180 nm CMOS imaging technology. A custom telescope with six MALTA planes has been developed for test beam campaigns at SPS, CERN, with the ability to host several devices under test. The telescope system has a dedicated custom readout, online monitoring integrated into DAQ with realtime hit map, time distribution and event hit multiplicity. It hosts a dedicated fully configurable trigger system enabling to trigger on coincidence between telescope planes and timing reference from a scintillator. The excellent time resolution performance allows for fast track reconstruction, due to the possibility to retain a low hit multiplicity per event which reduces the combinatorics. This paper reviews the architecture of the system and its performance during the 2021 and 2022 test beam campaign at the SPS North Area. [ABSTRACT FROM AUTHOR]

This paper introduces the R package clustAnalytics, which comprises a set of criteria for assessing the significance and stability of communities in networks found by any clustering algorithm. clustAnalytics works with graphs of class igraph from the R-package igraph, extended to handle weighted and/or directed graphs. clustAnalytics provides a set of community scoring functions, and methods to systematically compare their values to those of a suitable null model, which are of use when testing for cluster significance. It also provides a non parametric bootstrap method combined with similarity metrics derived from information theory and combinatorics, useful when testing for cluster stability, as well as a method to synthetically generate a weighted network with a ground truth community structure based on the preferential attachment model construction, producing networks with communities and scale-free degree distribution. [ABSTRACT FROM AUTHOR]

This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect 2, analogous to Richards's formula for defect 2 blocks of symmetric groups. By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding "q-decomposition numbers", i.e. the canonical basis coefficients in the level-1 q-deformed Fock space of type A^{(2)}_{2n}; a special case of a conjecture of Leclerc and Thibon asserts that these coefficients yield the spin decomposition numbers in characteristic 2n+1. Along the way, we prove some general results on q-decomposition numbers. This paper represents the first substantial progress on canonical bases in type A^{(2)}_{2n}. [ABSTRACT FROM AUTHOR]

This paper describes a new method of classification based on spectral analysis. The motivations behind developing the new model were the failures of the classical spectral cluster analysis based on combinatorial and normalized Laplacian for a set of real-world datasets of textual documents. Reasons of the failures are analysed. While the known methods are all based on usage of eigenvectors of graph Laplacians, a new classification method based on eigenvalues of graph Laplacians is proposed and studied. [ABSTRACT FROM AUTHOR]

In 1985 Crapo introduced in [3] a new mathematical object that he called geometry of circuits. Four years later, in 1989, Manin and Schechtman defined in [13] the same object and called it discriminantal arrangement, the name by which it is known now a days. Those discriminantal arrangements B(n,k,A 0 ) are builded from an arrangement A 0 of n hyperplanes in general position in a k-dimensional space and their combinatorics depends on the arrangement A 0 . On this basis, in 1997 Bayer and Brandt (see [2]) distinguished two different type of arrangements A 0 calling very generic the ones for which the intersection lattice of B(n,k,A 0 ) has maximum cardinality and non-very generic the others. Results on the combinatorics of B(n,k,A 0 ) in the very generic case already appear in Crapo [3] and in 1997 in Athanasiadis [1] while the first known result on non-very generic case is due to Libgober and the first author in 2018. In their paper [12] they provided a necessary and sufficient condition on A 0 for which the cardinality of rank 2 intersections in B(n,k,A 0 ) is not maximal anymore. In this paper we further develop their result providing a sufficient condition on A 0 for which the cardinality of rank r, r ≥ 2, intersections in B(n,k,A 0 ) decreases. [ABSTRACT FROM AUTHOR]

We investigate the structural constants of the KP hierarchy, which appear as universal coefficients in the paper of Natanzon–Zabrodin arXiv:1509.04472. It turns out that these constants have a combinatorial description in terms of transport coefficients in the theory of flow networks. Considering its properties we want to point out three novel directions of KP combinatorial structure research: connection with topological recursion, eigenvalue model for the structural constants and its deformations, possible deformations of KP hierarchy in terms of the structural constants. Firstly, in this paper we study the internal structure of these coefficients which involves: (1) construction of generating functions that have interesting properties by themselves; (2) restrictions on topological recursion initial data; (3) construction of integral representation or matrix model for these coefficients with non-trivial Ward identities. This shows that these coefficients appear in various problems of mathematical physics, which increases their value and significance. Secondly, we discuss their role in integrability of KP hierarchy considering possible deformation of these coefficients without changing the equations on τ -function. We consider several plausible deformations. While most failed even very basic checks, one deformation (involving Macdonald polynomials) passes all the simple checks and requires more thorough study. [ABSTRACT FROM AUTHOR]

A graph is called equimatchable if all of its maximal matchings have the same size. Lesk et al. [Equi-matchable graphs, Graph Theory and Combinatorics (Academic Press, London, 1984) 239–254] has provided a characterization of equimatchable bipartite graphs. Motivated by the fact that this characterization is not structural, Frendrup et al. [A note on equimatchable graphs, Australas. J. Combin. 46 (2010) 185–190] has also provided a structural characterization for equimatchable graphs with girth at least five, in particular, a characterization for equimatchable bipartite graphs with girth at least six. In this paper, we extend the characterization of Frendrup by eliminating the girth condition. For an equimatchable graph, an edge is said to be a critical-edge if the graph obtained by the removal of this edge is not equimatchable. An equimatchable graph is called edge-critical, denoted by ECE, if every edge is critical. Noting that each ECE-graph can be obtained from some equimatchable graph by recursively removing non-critical edges, each equimatchable graph can also be constructed from some ECE-graph by joining some non-adjacent vertices. Our study reduces the characterization of equimatchable bipartite graphs to the characterization of bipartite ECE-graphs. [ABSTRACT FROM AUTHOR]

The simple exclusion process studied in this paper consists of a system of K particles moving on the vertex set of a finite undirected connected graph. If there is one, the particle at x chooses one of the deg(x) neighbors of its current location uniformly at random at rate ρx > 0, and jumps to that vertex if and only if it is empty. Though this model is a natural mathematical object, it is also motivated by applications in the control of robotic swarms. After expressing the stationary distribution and the limiting occupation times of the process, we study how the total number of particles affects the occupation times ratios at different vertices. Using a novel qualitative argument based on combinatorial techniques, we prove that, while the occupation time at x increases with both D(x) = deg(x)=ρx and the total number of particles, the limiting occupation time increases faster with the total number of particles at vertices with a low D(x). [ABSTRACT FROM AUTHOR]

The notion of correlation immune functions has been introduced by Siegenthaler (1984) in symmetric cryptography in the framework of stream ciphers. At the conference CRYPTO’91 by Camion et al., it has been pointed out that this notion existed in statistics and combinatorics. It has recently been highlighted that such functions also play an important role in a new framework related to side-channel attack counter-measures. Since then, the interest in correlation immune Boolean functions has been renewed, and new challenges regarding these functions have appeared. Specifically, low Hamming weight correlation immune functions have been selected as useful for counter-measures to side-channel attacks. Despite their importance, the literature is not abundant in this research direction. Two very interesting articles in which such correlation immune functions were nicely explored, given this novel use of them. Carlet initiated the first one in 2013, and the second one is due to Carlet and Chen (2018). This paper deals with correlation immune Boolean functions aiming to produce more candidates of those processing low Hamming weights. We shall focus on correlation immune Boolean functions with Hamming weights power of 2 (which offer a flexibility to control the correlation immunity aspects) and present several methods of designing them. Some design methods are efficient and could be employed to derive such functions. Consequently, given two positive integers $n$ and $m$ , we derive new effective constructions of correlation immune Boolean functions with Hamming weight power of 2. Furthermore, an upper bound on the correlation immunity of the newly constructed $n$ -variable Boolean functions with Hamming weight $2^{m}$ was determined for $n-m\ge 0$. Besides, exact values and lower bounds on the maximum correlation immunity of those functions are explored and discussed, mainly when the values of $n$ and $m$ are very close. This paper also exhibits explicit examples of those correlation immune functions that illustrate our methods. [ABSTRACT FROM AUTHOR]

Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern–Fu–Tang and Heim–Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov–Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern–Fu–Tang conjecture and to show the Heim–Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions. [ABSTRACT FROM AUTHOR]

The inverse function x → x−1 on F2n, is one of the most studied functions in cryptography due to its widespread use as an S-box in block ciphers like AES. In this paper, we show that, if n ≥ 5, every function that is CCZ-equivalent to the inverse function is already EA-equivalent to it. This confirms a conjecture by Budaghyan, Calderini and Villa. We also prove that every permutation that is CCZ-equivalent to the inverse function is already affine equivalent to it. The majority of the paper is devoted to proving that there is no permutation polynomial of the form L1(x−1) + L2(x) over F2n if n ≥ 5, where L1,L2 are nonzero linear functions. In the proof, we combine Kloosterman sums, quadratic forms and tools from additive combinatorics. [ABSTRACT FROM AUTHOR]

A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this property, and this class has recently attracted much attention in complex dynamics. In the presence of escaping critical values, these curves break or split at critical points. In this paper, we develop combinatorial tools that allow us to provide a complete description of the escaping set of any criniferous function without asymptotic values on its Julia set. In particular, our description precisely reflects the splitting phenomenon. This combinatorial structure provides the foundation for further study of this class of functions. For example, we use these results in another paper to give the first full description of the topological dynamics of a class of transcendental entire maps with unbounded postsingular set. [ABSTRACT FROM AUTHOR]

Divide-and-conquer dividing by a half recurrences, of the form x n = a · x ⌈ n / 2 ⌉ + a · x ⌊ n / 2 ⌋ + p (n) , n ⩾ 2 , appear in many areas of applied mathematics, from the analysis of algorithms to the optimization of phylogenetic balance indices. These equations are usually "solved" by means of a Master Theorem that provides a bound for the growing order of xn, but not the solution's explicit expression. In this paper we give a finite explicit expression for this solution, in terms of the binary decomposition of n, when the independent term p(n) is a polynomial in ⌈n/2⌉ and ⌊n/2⌋. As an application, we obtain explicit formulas for several sequences of interest in phylogenetics, combinatorics, and computer science, for which no such formulas were known so far: for instance, for the Total Cophenetic index and the rooted Quartet index of the maximally balanced bifurcating phylogenetic trees with n leaves, and the sum of the bitwise AND operator applied to pairs of complementary numbers up to n. [ABSTRACT FROM AUTHOR]

In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals. [ABSTRACT FROM AUTHOR]

In this paper, we give a generalization of the Pascal triangle called the quasi s-Pascal triangle. For this, consider a set of lattice path, which is a dual approach to the definition of Ramirez and Sirvent: A Generalization of the k-bonacci Sequence from Riordan Arrays. The electronic journal of combinatorics, 22(1) (2015), 1-38. We give the recurrence relation for the sum of elements lying over finite ray of the quasi s-Pascal triangle, then, we establish a q-analogue of the coefficient of this triangle. Some identities are also given. [ABSTRACT FROM AUTHOR]

The article presents information on the 2005 Information Theory Society Paper Award, which is given annually to for a valuable publication in the field of information theory. This year the award was jointly awarded to researchers Shuo-Yen Robert Li, Raymond W. Yeung, and Ning Cai for their paper "Linear Network Coding" which appeared in the February 2003 issue of the journal "IEEE Transactions on Information Theory." Shuo-Yen Robert Li has done his B.S. in mathematics from National Taiwan University in 1970 and Ph.D. in mathematics from the University of California, Berkeley, in 1974. Yeung was born in Hong Kong on June 3, 1962. He has done B.S., M.Eng., Ph.D. in electrical engineering from Cornell University in Ithaca, New York. His research work include information theory and network coding. Ning Cai has done his B.S. in mathematics from the Normal College of Beijing in Beijing, China in 1982. He completed his Ph.D. in mathematics from the University of Bielefeld in Bielefeld, Germany in 1988. He has done research in the field of combinatorics, information theory and their applications.

Algebraic Nahm equations, considered in the paper, are polynomial equations, governing the q → 1 limit of the q-hypergeometric Nahm sums. They make an appearance in various fields: hyperbolic geometry, knot theory, quiver representation theory, topological strings and conformal field theory. In this paper we focus primarily on Nahm sums and Nahm equations that arise in relation with symmetric quivers. For a large class of them, we prove that quiver A-polynomials — specialized resultants of the Nahm equations, are tempered (the so-called K-theoretic condition). This implies that they are quantizable. Moreover, we find that their face polynomials obey a remarkable combinatorial pattern. We use the machinery of initial forms and mixed polyhedral decompositions to investigate the edges of the Newton polytope. We show that this condition holds for the diagonal quivers with adjacency matrix C = diag(α, α,... , α), α ≥ 2, and provide several checks for non-diagonal quivers. Our conjecture is that the K-theoretic condition holds for all symmetric quivers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a parametric type of Cauchy polynomials and study their characteristic and combinatorial properties. In particular, we show some determinant expressions. [ABSTRACT FROM AUTHOR]

Aiming at improving the accuracy of current tracking algorithms and tracking the target more quickly, a new target tracking method that is from multiscale analysis is presented, which is combined with combinatorial polygon matching. First, a target's range is estimated by analyzing multiple scales in the reference frame. Then, the pixels are judged as the target pixels within the target range determined in the previous step by calculating the relevant radial distance and judging the similarity of radial distance between target pixels to be detected and its background pixels. At the same time, the background modeling is updated according to the similarity between the incremental code of the background pixels and the corresponding pixels in the observed images. Last, the target trajectory is obtained by dynamic polygon matching. When encountering external interference, the performance of the improved algorithm is relatively stable. Compared with the existing methods, the proposed algorithm in this paper is verified by simulation, which is simple in operation and can track the target quickly and accurately. [ABSTRACT FROM AUTHOR]

Introduction/purpose: In this paper a new combinatorial proof of an already existing multiple sum with multiple binomial coefficients is given. The derived identity is related to the Fibonacci numbers. Methods: Combinatorial reasoning is used to obtain the results. Results: The already known identity was obtained by using a new combinatorial reasoning. Conclusions: The new combinatorial reasoning led to the solution of the already existing identity. [ABSTRACT FROM AUTHOR]

Symmetric functions play an important role in several subjects of mathematics, such as algebraic combinatorics, representation theory of finite groups and algebraic geometry. In this paper, we study the solutions of the following system of equations related to bases of symmetric functions: where , are natural numbers, and . Our main theorems generalize several known results. [ABSTRACT FROM AUTHOR]

In the paper [13] Păun, Polkowski and Skowron introduce several indiscernibility relations among strings that are infinite index equivalence or tolerance relations, and study lower and upper rough approximations of languages defined by them. In this paper we develop a further study of some of these indiscernibility relations among strings. We characterize the classes defined by them, and the rough approximations of general and context free languages under them. We also compare some of the rough approximations these relations produce to the ones given by the congruences defining testable, reverse testable, locally testable, piecewise testable and commutative languages. Those yield languages belonging to that families. Next, we modify some of the relations to obtain congruences, and study the families of languages the rough approximations under them give rise to. One of these modificated relations turns out to be the k-abelian congruence, that was defined by J. Karhumäki in [7], in the context of combinatorics on words. We show that it defines a pseudo-principal +-variety, a term defined in [9]. Our results in that work are then applied to determine when a given language has a best upper approximation in that family. Finally, we make some comments on the accuracy of the rough approximations obtained in each case. [ABSTRACT FROM AUTHOR]

Symetric designs are interesting objects of combinatorics, and have some relations with coding theory, difference sets, geometry and finite group theory. They have applications on statistics and design experiments. In the present paper we study an existing problem for symmetric design due to Bruck, Ryser and Chowla and write an algorithm by using their theorem called BRC Theorem. [ABSTRACT FROM AUTHOR]

This paper studies properties of binary runlength-limited sequences with additional constraints on their Hamming weight and/or their number of runs of identical symbols. An algebraic and a probabilistic (entropic) characterization of the exponential growth rate of the number of such sequences, i.e., their information capacity, are obtained by using the methods of multivariate analytic combinatorics, and properties of the capacity as a function of its parameters are stated. The second-order term in the asymptotic expansion of the rate of these sequences is also given, and the typical values of the relevant quantities are derived. Several applications of the results are illustrated, including bounds on codes for weight-preserving and run-preserving channels (e.g., the run-preserving insertion-deletion channel), a sphere-packing bound for channels with sparse error patterns, and the asymptotics of constant-weight sub-block constrained sequences. In addition, the asymptotics of a closely related notion— $q$ -ary sequences with fixed Manhattan weight—is briefly discussed, and an application in coding for molecular timing channels is illustrated. [ABSTRACT FROM AUTHOR]

Let H = (V (H) , E (H)) be a hypergraph, where V (H) is a set of vertices and E (H) is a set of non-empty subsets of V (H) called edges. If all edges of H have the same cardinality r , then H is an r -uniform hypergraph. A hypergraph H ′ = (V (H ′) , E (H ′)) is called a hypersubgraph of a hypergraph H = (V (H) , E (H)) if V (H ′) ⊆ V (H) and E (H ′) ⊆ E (H). The v e r t e x - c o n n e c t i v i t y of hypergraph H , denoted by κ (H) , is the cardinality of a minimum vertex set S such that H − S is disconnected or is a trival hypergraph. We call H k - c o n n e c t e d if κ (H) ≥ k. Tian, Lai and Meng [Y. Z. Tian, H.-J. Lai, J. X. Meng, On the sizes of vertex- k -maximal r -uniform hypergraphs, Graphs and Combinatorics 35(5) (2019) 1001–1010] conjectered that, for sufficiently large n , every n -vertex r -uniform hypergraph with no (k + 1) -connected hypersubgraphs has at most n r − n − k r + n k − 2 k r edges. This upper bound is equal to 1 2 k (n − k) 2 + 2 3 k 2 (n − k) − 3 2 k − 2 9 (n − k) when r = 3. In this paper, we prove that for k ≥ 3 and n ≥ 2. 6 7 k , every n -vertex 3 -uniform hypergraph with no (k + 1) -connected hypersubgraphs has at most 3 5 k (n − k) 2 + 1 2 k 2 (n − k) − k (n − k) + 1 3 (n − k) edges. [ABSTRACT FROM AUTHOR]

In this paper, we delve into studying some relations between the structure of the cycles and spanning trees of a graph through the lens of its cycle and spanning tree hypergraphs which are hypergraphs with the edge set of the graph as their vertices and the edge sets of the cycles and spanning trees as their hyperedges respectively. In particular, we investigate relations between these hypergraphs from the perspective of the VC-dimension and some important separating and covering features of hypergraph theory and amongst the results, for example show that the VC-dimension of the cycle hypergraph is less than or equal to the VC-dimension of the spanning tree hypergraph and their gap can be arbitrary large. Note that VC-dimension is an important measure of complexity and a fundamental notion in numerous fields such as extremal combinatorics, graph theory, statistics and the theory of machine learning. Also we compare the separating and covering features of the mentioned hypergraphs and for instance show that the separating number of the cycle hypergraph is less than or equal to that of the spanning tree hypergraph. These hypergraphs help us to make several connections between cycles and spanning trees of graphs and compare their complexities. [ABSTRACT FROM AUTHOR]

This study develops a combinatorial approach for nonparametric short-term queue length estimation in terms of cycle-by-cycle partially observed queues from probe vehicles (PV). The method does not assume random arrivals and does not assume any primary parameters or estimation of any parameters but uses simple algebraic expressions that only depend on signal timing. For an approach lane at a traffic intersection, the conditional queue lengths given probe vehicle location, count, time, and analysis interval (e.g., at the end of the red signal phase) are represented by a Negative Hypergeometric distribution. The simple analytical estimators obtained are compared with parametric methods from literature and highway capacity manual methods using field test data and simulation data involving probe vehicles. The analysis indicates that the nonparametric models presented in this paper match the accuracy of the parametric ones used in the field test and simulated data for estimating queue lengths. • Two new models are derived for short-term queue length estimation. • The models are based on combinatorics and nonparametric. • The models are closed-form input–output formulas. • New probability mass functions for traffic queues are derived. • The models can be used in connected traffic signals. [ABSTRACT FROM AUTHOR]

Let (G,V) be a prehomogeneous vector space, let O be any G(Fq)-invariant subset of V (Fq), and let F be the characteristic function of O. In this paper we develop a method for explicitly and efficiently evaluating the Fourier transform ΦF, based on combinatorics and linear algebra. We then carry out these computations in full for each of five prehomogeneous vector spaces, including the 12-dimensional space of pairs of ternary quadratic forms. Our computations reveal that these Fourier transforms enjoy a great deal of structure, and sometimes exhibit more than square root cancellation on average. These Fourier transforms naturally arise in analytic number theory, where explicit formulas (or upper bounds) lead to sieve level of distribution results for related arithmetic sequences. We describe some examples, and in a companion paper we develop a new method to do so, designed to exploit the particular structure of these Fourier transforms. [ABSTRACT FROM AUTHOR]

For a graph G , the tree Kneser graph KG (G , T t) has all tree subgraphs of G with t vertices as vertex set and two vertices are adjacent if their corresponding trees are edge-disjoint. Also, the r -th cut number of G is the minimum number of edges between the parts of a partition of the vertex set of G into two parts such that each part has size at least r. In this paper, we investigate the chromatic number of tree Kneser graphs. Roughly speaking, we prove that for any nonnegative integer function t = t (n) , where n − t = o (n) , if n is sufficiently large, then for any dense graph G with n vertices, the chromatic number of the tree Kneser graph KG (G , T t) is equal to the (n − t + 1) -th cut number of G. In particular, as a consequence of this result, if n is large enough, then for any dense graph G with n vertices, the chromatic number of the tree Kneser graph KG (G , T n) is equal to the minimum cut (minimum degree) of G. This result can be somehow considered a reminiscent of the Nash-Williams disjoint trees theorem. The proof is based on a lower bound for the chromatic number of graphs found by the present authors Alishahi and Hajiabolhassan (2015) [1] which is inspired by Tucker's lemma, an equivalent combinatorial version of the Borsuk-Ulam theorem. Finally, motivated by the aforementioned results, we close the paper by a discussion on the complexity of determining the r -th cut number of graphs. [ABSTRACT FROM AUTHOR]

In the papers (Benoumhani 1996;1997), Benoumhani defined two polynomials F m , n , 1 (x) and F m , n , 2 (x). Then, he defined A m (n , k) and B m (n , k) to be the polynomials satisfying F m , n , 1 (x) = ∑ k = 0 n A m (n , k) x n − k (x + 1) k and F m , n , 1 (x) = ∑ k = 0 n B m (n , k) x n − k (x + 1) k. In this paper, we give a combinatorial interpretation of the coefficients of A m + 1 (n , k) and prove a symmetry of the coefficients, i.e., [ m s ] A m + 1 (n , k) = [ m n − s ] A m + 1 (n , n − k). We give a combinatorial interpretation of B m + 1 (n , k) and prove that B m + 1 (n , n − 1) is a polynomial in m with non-negative integer coefficients. We also prove that if n ≥ 6 then all coefficients of B m + 1 (n , n − 2) except the coefficient of m n − 1 are non-negative integers. For all n , the coefficient of m n − 1 in B m + 1 (n , n − 2) is − (n − 1) , and when n ≤ 5 some other coefficients of B m + 1 (n , n − 2) are also negative. [ABSTRACT FROM AUTHOR]

We continue the study of a class of signed graphs called finite connected loop-free edge-bipartite graphs Δ (bigraphs, for short), started in Simson (2013) [33] and continued in Gąsiorek et al. (2016) [14] and Simson-Zając (2017) [38]. In this paper we present an algorithmic approach to the study of non-negative bigraphs Δ with n + r ≥ 2 vertices of corank r ≥ 1 , that is, bigraphs with the symmetric Gram matrix G Δ : = 1 2 ( G ˇ Δ + G ˇ Δ t r) ∈ M n + r (1 2 Z) positive semi-definite of rank n ≥ 1 , where G ˇ Δ ∈ M n + r (Z) is the non-symmetric Gram matrix of Δ. One of the main results of this paper are Theorems 2.1 and 2.17 asserting that every such a bigraph Δ with n + r ≥ 2 vertices can be obtained from a corank zero (i.e. positive) connected loop-free bigraph Δ ′ with n ≥ 1 vertices by an r -point extension procedure (Δ ′ , u (1) , ... , u (r)) ↦ Δ : = Δ ′ [ [ u (1) , ... , u (r) ] ] along the r ≥ 1 roots u (1) , ... , u (r) ∈ Z n of the positive definite Gram form q Δ ′ : Z n → Z , v ↦ v G ˇ Δ ′ v t r. It is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop-free non-negative corank r ≥ 1 bigraphs with n + r ≥ 2 vertices, from the corank-zero bigraphs Δ ′ with n ≥ 1 vertices. [ABSTRACT FROM AUTHOR]

The cyclic shift graph of a monoid is the graph whose vertices are elements of the monoid and whose edges link elements that differ by a cyclic shift. This paper examines the cyclic shift graphs of 'plactic-like' monoids, whose elements can be viewed as combinatorial objects of some type: aside from the plactic monoid itself (the monoid of Young tableaux), examples include the hypoplactic monoid (quasi-ribbon tableaux), the sylvester monoid (binary search trees), the stalactic monoid (stalactic tableaux), the taiga monoid (binary search trees with multiplicities), and the Baxter monoid (pairs of twin binary search trees). It was already known that for many of these monoids, connected components of the cyclic shift graph consist of elements that have the same evaluation (that is, contain the same number of each generating symbol). This paper focuses on the maximum diameter of a connected component of the cyclic shift graph of these monoids in the rank- n case. For the hypoplactic monoid, this is n − 1 ; for the sylvester and taiga monoids, at least n − 1 and at most n ; for the stalactic monoid, 3 (except for ranks 1 and 2, when it is respectively 0 and 1); for the plactic monoid, at least n − 1 and at most 2 n − 3. The current state of knowledge, including new and previously-known results, is summarized in a table. [ABSTRACT FROM AUTHOR]

After fixing a canonical ordering (or labeling) of the elements of a finite poset, one can associate each linear extension of the poset with a permutation. Some recent papers consider specific families of posets and ask how many linear extensions give rise to permutations that avoid certain patterns. We build off of two of these papers. We first consider pattern avoidance in k-ary heaps, where we obtain a general result that proves a conjecture of Levin, Pudwell, Riehl, and Sandberg in a special case. We then prove some conjectures that Anderson, Egge, Riehl, Ryan, Steinke, and Vaughan made about pattern-avoiding linear extensions of rectangular posets. [ABSTRACT FROM AUTHOR]