Given an F-represented matroid (M,ρ) with the ground set [m], the representation ρ naturally defines a hyperplane arrangement Aρ. We will study its parallel translates Aρ,g of Aρ for all g∈Fm. Its intersection semi-lattices L(Aρ,g) and the characteristic polynomials χ(Aρ,g,t) will be classified by the intersection lattice of the derived arrangement Aδρ, which is a hyperplane arrangement associated

In this paper, we study the periodicity structure of finite field linear recurring sequences whose period is not necessarily maximal and determine necessary and sufficient conditions for the characteristic polynomial f to have exactly two periods in the sense that the period of any sequence generated by f is either one or a unique integer greater than one.

We prove a conjecture of Thomas Lam that the face posets of stratified spaces of planar resistor networks are shellable. These posets are called uncrossing partial orders. This shellability result combines with Lam's previous result that these same posets are Eulerian to imply that they are CW posets, namely that they are face posets of regular CW complexes. Certain subposets of uncrossing partial

We call weak Markoff theory the theory of Markoff restricted to integral quadratic forms (instead of real ones), and to quadratic real numbers (instead of general real numbers). We show that weak Markoff theory may be reduced to combinatorial properties of Christoffel words, avoiding the use of bi-infinite sequences. These properties are two new characterizations of Christoffel words, one using the

We propose a unified approach to prove general formulas for the joint distribution of an Eulerian and a Mahonian statistic over a set of colored permutations by specializing Poirier's colored quasisymmetric functions. We apply this method to derive formulas for Euler–Mahonian distributions on colored permutations, derangements and involutions. A number of known formulas are recovered as special cases

In this paper, we first establish a version of the central limit theorem for a double sequence {pi(n)} that satisfies a linear recurrence relation. Then we find and prove that under some commonly observed conditions, the sequence of embedding distributions of an H-linear family of graphs with spiders is asymptotic to a normal distribution. Applications are given to some well-known path-like and ring-like

In this paper, first steps are taken towards characterizing rank-decorated lattices of cyclic flats Z(M) that belong to matroids M that can be represented over a prescribed finite field Fq. Two natural maps from Z(M) to the lattice of cyclic flats of a minor of M are given. Binary matroids are characterized via their lattice of cyclic flats. It is shown that the lattice of cyclic flats of a simple

This contribution gives an extensive study on the Wiener indices, the number of closed walks, the coefficients of some graph polynomials (the adjacency polynomial, the Laplacian polynomial, the edge cover polynomial and the independence polynomial) of trees. Csikvári (2010) [4] introduced the generalized tree shift, which keeps the number of vertices of trees. Applying the generalized tree shifts and

A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian; this baseline result is used as the basis of existence proofs for universal cycles (also known as ucycles or generalized deBruijn cycles or U-cycles) of several combinatorial objects. The existence of ucycles is often dependent on the specific representation that we use for the combinatorial objects. For example

A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al. in 2015. In a recent paper Kitaev and Zhang examined the distribution of the aforementioned patterns. The aim of this paper is to prove more equidistributions of mesh pattern and confirm Kitaev and Zhang's four conjectures by constructing two involutions on permutations.

In the framework of the probabilistic method in combinatorics, we provide a systematization of the entropy compression method clarifying the setting in which it can be applied and providing a theorem yielding a general constructive criterion. We finally elucidate, through topical examples, the effectiveness of the entropy-compression criterion in comparison with the Lovász Local Lemma criterion and

This paper studies subsets of one-sided shift spaces on a finite alphabet. Such subsets arise in symbolic dynamics, in fractal constructions, and in number theory. We study a family of decimation operations, which extract subsequences of symbol sequences in infinite arithmetic progressions, and show these operations are closed under composition. We also study a family of n-ary interleaving operations

The NC versus P-hard classification of the prediction problem for sandpiles on the two dimensional grid with von Neumann neighborhood is a famous open problem. In this paper we make two kinds of progresses, by studying its freezing variant. First, it enables to establish strong connections with other well known prediction problems on networks of threshold Boolean functions such as majority. Second

In his 1998 Fubini Lectures, Rota discusses twelve problems in probability that “no one likes to bring up”. The first problem calls for a revision of the notion of a sample space, guided by the belief that mention of sample points in a probabilistic argument is bad form and that a “pointless” foundation of probability should be provided by algebras of random variables. In 1958 Chang introduced MV-algebras

We provide an explicit product formula for the number of spanning trees of the Bruhat graph of the symmetric group, that is, the graph where two permutations π and σ are connected with an edge if πσ−1 is a transposition. We also give the number of spanning trees for the graph where the two permutations are connected if πσ−1 is an r-cycle for r even. For r odd we obtain the similar result for the alternating

We consider numeration systems based on a d-tuple U=(U1,…,Ud) of sequences of integers and we define (U,K)-regular sequences through K-recognizable formal series, where K is any semiring. We show that, for any d-tuple U of Pisot numeration systems and any semiring K, this definition does not depend on the greediness of the U-representations of integers. The proof is constructive and is based on the

Our aim in the present work is to identify all the possible standing wave configurations involving few vortices of different charges in an atomic Bose-Einstein condensate (BEC). In this effort, we deploy the use of a computational algebra approach in order to identify stationary multi-vortex states with up to 6 vortices. The use of invariants and symmetries enables deducing a set of equations in elementary

In this paper, we show that the joint distribution of descents and major indices in conjugacy class is asymptotically bivariate normal. This generalizes the authors' previous work on the asymptotical normality of descents in conjugacy classes, where the asymptotic parameters depended only on the density of fixed points. The result is achieved by two key ingredients; one is a variation of the continuity

Chvátal's conjecture on the intersecting family of the faces of the simplicial complex is a long-standing problem in combinatorics. Snevily gave an affirmative answer to this conjecture for near-cone complex. Woodroofe gave Erdős-Ko-Rado type theorem for near-cone complex by using algebraic shift method. Motivated by these results, we concern with the restricted intersecting family for the simplicial

We study the distributional properties of horizontal visibility graphs associated with random restrictive growth sequences and random set partitions of size n. Our main results are formulas expressing the expected degree of graph nodes in terms of simple explicit functions of a finite collection of Stirling and Bernoulli numbers.

In this paper we consider a general class E of self-similar sets with complete overlaps. Given a self-similar iterated function system Φ=(E,{fi}i=1m)∈E on the real line, for each point x∈E we can find a sequence (ik)=i1i2…∈{1,…,m}N, called a coding of x, such thatx=limn→∞⁡fi1∘fi2∘⋯∘fin(0). For k=1,2,…,ℵ0 or 2ℵ0 we investigate the subset Uk(Φ) which consists of all x∈E having precisely k different codings

This paper provides a mathematical approach to the study of chromatic aberration in metalenses. It is shown that radiation of a given wavelength is refracted according to a generalized Snell's law, which together with the notion of envelope, yields the existence of phase discontinuities. This is then used to establish a quantitative measure of dispersion in metalenses, concluding that in the visible

Szemerédi's regularity lemma is one instance in a family of regularity lemmas, replacing the definition of density of a graph by a more general coefficient. Recently, Fan Chung proved another instance, a regularity lemma for clustering graphs, and asked whether good upper bounds could be derived for the quantitative estimates it supplies. We answer this question in the negative, for every generalized

We introduce a notion of dimension for the solution set of a system of algebraic difference equations that measures the degrees of freedom when determining a solution in the ring of sequences. This number need not be an integer, but, as we show, it satisfies properties suitable for a notion of dimension. We also show that the dimension of a difference monomial is given by the covering density of its

Whitney's 2-Isomorphism Theorem characterises when two graphs have isomorphic cycle matroids. We present an analogue of this theorem for graphs embedded in surfaces by characterising when two graphs in surface have isomorphic delta-matroids.

Interval parking functions (IPFs) are a generalization of ordinary parking functions in which each car is willing to park only in a fixed interval of spaces. Each interval parking function can be expressed as a pair (a,b), where a is a parking function and b is a dual parking function. We say that a pair of permutations (x,y) is reachable if there is an IPF (a,b) such that x,y are the outcomes of a

A finite simple graph Γ determines a quotient PΓ of the pure braid group, called a graphic arrangement group. We analyze homomorphisms of these groups defined by deletion of sets of vertices, using methods developed in prior joint work with R. Randell. We show that, for a K4-free graph Γ, a product of deletion maps is injective, embedding PΓ in a product of free groups. Then PΓ is residually free,

The chromatic polynomial of a graph G, denoted P(G,m), is equal to the number of proper m-colorings of G. The list color function of graph G, denoted Pℓ(G,m), is a list analogue of the chromatic polynomial that has been studied since 1992, primarily through comparisons with the corresponding chromatic polynomial. DP-coloring (also called correspondence coloring) is a generalization of list coloring

A method is presented for computing the form of highest degree of the implicit equation of a rational surface, defined by means of a rational parametrization. Determining the form of highest degree is useful to study the asymptotic behavior of the surface, to perform surface recognition, or to study symmetries of surfaces, among other applications. The method is efficient, and works generally better

We give a combinatorial description of the toric ideal of invariants of the Cavender-Farris-Neyman model with a molecular clock (CFN-MC) on a rooted binary phylogenetic tree and prove results about the polytope associated to this toric ideal. Key results about the polyhedral structure include that the number of vertices of this polytope is a Fibonacci number, the facets of the polytope can be described

We provide bivariate asymptotics for the poly-Bernoulli numbers, a combinatorial array that enumerates lonesum matrices, using the methods of Analytic Combinatorics in Several Variables (ACSV). For the diagonal asymptotic (i.e., for the special case of square lonesum matrices) we present an alternative proof based on Parseval's identity. In addition, we provide an application in Algebraic Statistics

W. Gosper in 2001 introduced a constant Πq, and conjectured many interesting identities on this constant. In this paper, applying the theory of modular forms, we prove one of these Πq-identities.

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias et al. (2016) [4]. Let Um,d denote the uniform matroid of rank d on a set of m+d elements. Gedeon et al. (2017) [7] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of Um,d using equivariant Kazhdan-Lusztig polynomials. In this paper we give an alternative explicit formula, which allows us to

Sharp complex Lp affine isoperimetric inequalities are established for the entire class of complex Lp projection bodies and the entire class of complex Lp moment bodies.

We discuss certain models of irreducible representations of Lie algebra sl(2,C) from the special matrix functions point of view. These models are constructed in terms of matrix differential operators and matrix difference differential operators and are connected through a matrix integral transformation. In the process, we find new matrix function identities involving one and two variable special matrix

Given a permutation w, we look at the range of how often a simple reflection σk appears in reduced decompositions of w. We compute the minimum and give a sharp upper bound on the maximum. That bound is in terms of 321- and 3412-patterns in w, specifically as they relate in value and position to k. We also characterize when that minimum and maximum are equal, refining a previous result that braid moves

We study a family of sequences cn(a2,…,ar), where r≥2 and a2,…,ar are real parameters. We find a sufficient condition for positive definiteness of the sequence cn(a2,…,ar) and check several examples from OEIS. We also study relations of these sequences with the free and monotone convolution.

Fici, Restivo, Silva, and Zamboni define a k-anti-power to be a concatenation of k consecutive words that are pairwise distinct and have the same length. They ask for the maximum k such that every aperiodic recurrent word must contain a k-anti-power, and they prove that this maximum must be 3, 4, or 5. We resolve this question by demonstrating that the maximum is 5. We also conjecture that if W is

We show that the number of signed permutations avoiding 1234 equals the number of signed permutations avoiding 2143 (also called vexillary signed permutations), resolving a conjecture by Anderson and Fulton. The main tool that we use is the generating tree developed by West. Many further directions are mentioned in the end.

Given a positive integer M and a real number x>0, let U(x) be the set of all bases q∈(1,M+1] for which there exists a unique sequence (di)=d1d2… with each digit di∈{0,1,…,M} satisfyingx=∑i=1∞diqi. The sequence (di) is called a q-expansion of x. In this paper we investigate the local dimension of U(x) and prove a ‘variation principle’ for unique non-integer base expansions. We also determine the critical

In stochastic analysis, a standard method to study a path is to work with its signature. This is a sequence of tensors of different order that encode information of the path in a compact form. When the path varies, such signatures parametrize an algebraic variety in the tensor space. The study of these signature varieties builds a bridge between algebraic geometry and stochastics, and allows a fruitful

In this paper we introduce abstract string modules and give an explicit bijection between the submodule lattice of an abstract string module and the perfect matching lattice of the corresponding abstract snake graph. In particular, we make explicit the direct correspondence between a submodule of a string module and the perfect matching of the corresponding snake graph. For every string module we define

Nevanlinna-Herglotz functions play a fundamental role for the study of infinitely divisible distributions in free probability [11]. In the present paper we study the role of the tangent function, which is a fundamental Herglotz-Nevanlinna function [28], [23], [54], and related functions in free probability. To be specific, we show that the functiontan⁡z1−xtan⁡z of Carlitz and Scoville [17, (1.6)] describes

We analyze the complexity of fitting a variety, coming from a class of varieties, to a configuration of points in Rn. The complexity measure, called the algebraic complexity, computes the Euclidean Distance Degree (EDdegree) of a certain variety called the hypothesis variety as the number of points in the configuration increases. Finally, we establish a connection to complexity of architectures of

Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for “nearest” point problems of the algebraic variety. It is well known that the latter is an upper bound for the former. While this bound may be tight, many varieties appearing in optimization, engineering

We study a new notion of cyclic avoidance of abelian powers. A finite word w avoids abelian N-powers cyclically if for each abelian N-power of period m occurring in the infinite word wω, we have m≥|w|. Let A(k) be the least integer N such that for all n there exists a word of length n over a k-letter alphabet that avoids abelian N-powers cyclically. Let A∞(k) be the least integer N such that there

Let N0(m,n) be the number of odd Durfee symbols of n with odd rank m, and N0(a,M;n) be the number of odd Durfee symbols of n with odd rank congruent to a modulo M. We give explicit formulas for the generating functions of N0(a,M;n) and their ℓ-dissections where 0≤a≤M−1 and M,ℓ∈{2,4,8}. From these formulas, we obtain some interesting arithmetic properties of N0(a,M;n). Furthermore, let Dk0(n) denote

In this paper, we propose an (s+d,d)-abacus for (s,s+d,…,s+pd)-core partitions and establish a bijection between the (s,s+d,…,s+pd)-core partitions and the rational Motzkin paths of type (s+d,−d). This result not only gives a lattice path interpretation of the (s,s+d,…,s+pd)-core partitions but also counts them with an explicit formula. Also we enumerate (s,s+1,…,s+p)-core partitions with k corners

To any n×n Latin square L, we may associate a unique sequence of mutually orthogonal permutation matrices P=P1,P2,...,Pn such that L=L(P)=∑kPk. Brualdi and Dahl (2018) described a generalisation of a Latin square, called an alternating sign hypermatrix Latin-like square (ASHL), by replacing P with an alternating sign hypermatrix (ASHM). An ASHM is an n×n×n (0,1,-1)-hypermatrix in which the non-zero

Consider a curve Γ in a domain D in the plane R2. Thinking of D as a piece of paper, one can make a curved folding P in the Euclidean space R3. The singular set C of P as a space curve is called the crease of P and the initially given plane curve Γ is called the crease pattern of P. In this paper, we show that in general there are four distinct non-congruent curved foldings with a given pair consisting

A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle peaks and excedances of permutations. We prove a series of new general formulae expressing polynomials counting permutations by various excedance statistics in terms

In 1997, Shor and Laflamme defined weight enumerators for quantum error-correcting codes and derived a MacWilliams identity. Recently, we proposed two new notions, the double weight enumerators and complete weight enumerators, for binary quantum codes. Now we will extend these results to nonbinary quantum codes. Based on the MacWilliams identities for these enumerators, we explicitly determine the

This article is mainly devoted to studying a novel geometric & nonlinear capacity induced by the affine bounded variation in the Euclidean space Rn which is truly distinct from the classical BV-capacity in dimension n>1.

It is well known that the monomer-dimer problem is very interesting but difficult in statistical physics, which is equivalent to the enumerative problem of matchings of a graph in combinatorics. In this paper, using the generating function of edge subsets of a graph, we obtain the solution of the monomer-dimer problem of a number of graphs.

We study the symmetric function and polynomial combinatorial invariants of Hopf algebras of permutations, posets and graphs. We investigate their properties and the relations among them. In particular, we show that the chromatic symmetric function and many other invariants have a property we call positively h-alternating. This property of positively h-alternating leads to Schur positivity and e-positivity

In this paper, we introduce new concept of (q,c)-derivative operator of an analytic function, which generalizes the ordinary q-derivative operator. From this definition, we give the concept of (q,c)-Rogers-Szegö polynomials, and obtain the expanded theorem involving (q,c)-Rogers-Szegö polynomials. In addition, we construct two kinds (q,c)-exponential operators, apply them to (q,c)-exponential functions

Consider a Poisson point process within a convex set in a Euclidean space. The Vietoris-Rips complex is the clique complex over the graph connecting all pairs of points with distance at most δ. Summing powers of the volume of all k-dimensional faces defines the volume-power functionals of these random simplicial complexes. The asymptotic behavior of the volume-power functionals of the Vietoris-Rips

By applying the Chinese remainder theorem for coprime polynomials and the “creative microscoping” method recently introduced by the author and Zudilin, we establish parametric generalizations of three q-supercongruences modulo the fourth power of a cyclotomic polynomial. The original q-supercongruences then follow from these parametric generalizations by taking the limits as the parameter tends to

A phylogenetic network is a graph-theoretical tool that is used by biologists to represent the evolutionary history of a collection of species. One potential way of constructing such networks is via a distance-based approach, where one is asked to find a phylogenetic network that in some way represents a given distance matrix, which gives information on the evolutionary distances between present-day

We introduce a construction which allows us to identify the elements of the skeletons of a CW-complex P(m) and the monomials in m variables. From this, we infer that there is a bijection between finite CW-subcomplexes of P(m), which are quotients of finite simplicial complexes, and certain bigraded standard Artinian Gorenstein algebras, generalizing previous constructions of Faridi and ourselves. We