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

We consider the problem of selecting an n-member committee made up of one of m candidates from each of n distinct departments. Using an algebraic approach, we analyze positional voting procedures, including the Borda count, as QSm≀Sn-module homomorphisms. In particular, we decompose the spaces of voter preferences and election results into simple QSm≀Sn-submodules and apply Schur's Lemma to determine

Function classes are collections of Boolean functions on a finite set, which are fundamental objects of study in theoretical computer science. We study algebraic properties of ideals associated to function classes previously defined by the third author. We consider the broad family of intersection-closed function classes, and describe cellular free resolutions of their ideals by order complexes of

The Billera-Holmes-Vogtmann (BHV) space of weighted trees can be embedded in Euclidean space, but the extrinsic Euclidean mean often lies outside of treespace. Sturm showed that the intrinsic Fréchet mean exists and is unique in treespace. This Fréchet mean can be approximated with an iterative algorithm, but bounds on the convergence of the algorithm are not known, and there is no other known polynomial

In this article, we study the Bruhat-Chevalley-Renner order on the complex symplectic monoid MSpn. After showing that this order is completely determined by the Bruhat-Chevalley-Renner order on the linear algebraic monoid of n×n matrices Mn, we focus on the Borel submonoid of MSpn. By using this submonoid, we introduce a new set of type B set partitions. We determine their count by using the “folding”

Given any numeration system, we call carry propagation at a number N the number of digits that are changed when going from the representation of N to the one of N+1, and amortized carry propagation the limit of the mean of the carry propagations at the first N integers, when N tends to infinity, if this limit exists. In the case of the usual base p numeration system, it can be shown that the limit

We address the classical factorization problem of a one dimensional Schrödinger operator −∂2+u−λ, for a stationary potential u of the KdV hierarchy but, in this occasion, a “parameter” λ is considered. Inspired by the more effective approach of Gesztesy and Holden to the “direct” spectral problem in [14], we give a symbolic algorithm by means of differential elimination tools to achieve the aimed factorization

Let A be a Weyl arrangement. We introduce and study the notion of A-Eulerian polynomial producing an Eulerian-like polynomial for any subarrangement of A. This polynomial together with shift operator describes how the characteristic quasi-polynomial of a new class of arrangements containing ideal subarrangements of A can be expressed in terms of the Ehrhart quasi-polynomial of the fundamental alcove

We show that polynomials associated with automatic sequences satisfy a certain recurrence relation when evaluated at a root of unity. This generalizes a result of Brillhart, Lomont and Morton on the Rudin–Shapiro polynomials. We study the order of such a relation and the integrality of its coefficients.

Consider a cyclically ordered collection of r equi-numerous “agent” sets with strict preferences of every agent over the agents from the next set. A weakly stable cyclic matching is a partition of the set of agents into disjoint union of r-long cycles, one agent from each set per cycle, such that there are no destabilizing r-long cycles, i.e. cycles in which every agent strictly prefers its successor

Let r be a positive integer and let Gn be the reflection group of n×n monomial matrices whose entries are rth complex roots of unity and let k≤n. We define and study two new graded quotients Rn,k and Sn,k of the polynomial ring C[x1,…,xn] in n variables. When k=n, both of these quotients coincide with the classical coinvariant algebra attached to Gn. The algebraic properties of our quotients are governed

We consider polygonal tilings of certain regions and use these to give intuitive definitions of tiling-based perimeter and area. We apply these definitions to rhombic tilings of Elnitsky polygons, computing sharp bounds and average values for perimeter tiles in convex centrally symmetric 2n-gons. These bounds and values have implications for the combinatorics of reduced decompositions of permutations

The so-called binary perfect phylogeny with persistent characters has recently been thoroughly studied in computational biology as it is less restrictive than the well-known binary perfect phylogeny. Here, we focus on the notion of (binary) persistent characters, i.e. characters that can be realized on a phylogenetic tree by at most one 0→1 transition followed by at most one 1→0 transition in the tree

Given a sequence of s-uniform hypergraphs {Hn}n≥1, denote by Tp(Hn) the number of edges in the random induced hypergraph obtained by including every vertex in Hn independently with probability p∈(0,1). Recent advances in the large deviations of low complexity non-linear functions of independent Bernoulli variables can be used to show that tail probabilities of Tp(Hn) are precisely approximated by the

We study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of gln restricted to the dual fundamental Weyl chamber. We obtain generating functions that count flats and faces of a given dimension. This counting is interpreted in physics as the enumeration of the phases of the Coulomb and mixed Coulomb-Higgs

Classical Gončarov polynomials arose in numerical analysis as a basis for the solutions of the Gončarov interpolation problem. These polynomials provide a natural algebraic tool in the enumerative theory of parking functions. By replacing the differentiation operator with a delta operator and using the theory of finite operator calculus, Lorentz, Tringali and Yan introduced the sequence of generalized

We study the polyhedral geometry of the hyperplanes orthogonal to the weights of the first and the second fundamental representations of sln inside the dual fundamental Weyl chamber. We obtain generating functions that enumerate the flats and the faces of a fixed dimension. In addition, we describe the extreme rays of the incidence geometry and classify simplicial faces. From the perspective of supersymmetric

In 2004, Ehrenfeucht, Harju, and Rozenberg showed that any graph on a vertex set V can be obtained from a complete graph on V via a sequence of the operations of complementation, switching edges and non-edges at a vertex, and local complementation. The last operation involves taking the complement in the neighbourhood of a vertex. In this paper, we consider natural generalizations of these operations

We consider a finite field model of the X-ray transform that integrates functions along lines in dimension 3, within the context of finite fields. The admissibility problem asks for minimal sets of lines for which the restricted transform is invertible. Graph theoretic conditions are known which characterize admissible collections of lines, and these have been counted using a brute force computer program

Let A be a set of vertices in a graph G. An A-path is a non-trivial path in G that has both of its ends in A. In 1961, Gallai showed that, for any integer k, either G has k disjoint A-paths or there is a set of at most 2(k−1) vertices that hits all of the A-paths. There have been a number of extensions of this result; in each of these extensions we want to find a maximum collection of disjoint “allowable”

We give a new proof of Rado's Theorem that, given a partition (X1,…,Xn) of the ground set of a matroid M, there is an independent set of M containing exactly one element from each of the sets (X1,…,Xn) if and only if for each subset J of {1,…,n} we have r(∪j∈JXj)≥|J|.

A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable matroids is strongly fractal. We prove that the class of sparse paving matroids with at most k circuit-hyperplanes is a strongly fractal class when k is at least three