Mock theta functions can be represented by Eulerian forms, Hecke-type double sums, Appell–Lerch sums, and Fourier coefficients of meromorphic Jacobi forms. In view of some q-series identities, we establish three two-parameter mock theta functions, and express them in terms of Appell-Lerch sums. In addition, we find three mock theta functions in Eulerian forms. Then in light of their Hecke-type double

There exists a well established differential topological theory of singularities of ordinary differential equations. It has mainly studied scalar equations of low order. We propose an extension of the key concepts to arbitrary systems of ordinary or partial differential equations. Furthermore, we show how a combination of this geometric theory with (differential) algebraic tools allows us to make parts

For a fixed set X containing n taxon labels, an ordered pair consisting of a gene tree topology G and a species tree topology S bijectively labeled with the labels of X possesses a set of coalescent histories—mappings from the set of internal nodes of G to the set of edges of S describing possible lists of edges in S on which the coalescences in G take place. Enumerations of coalescent histories for

In this paper we consider matrix codes equipped with either the Hamming or the rank distance and study the optimal sizes of such codes with a given minimum distance. We use a maximality argument (the Gilbert-Varshamov argument) to show that, asymptotically, maximum size codes are optimal and have the same size for both rank and Hamming distances.

Recently, the perfect phylogeny model with persistent characters has attracted great attention in the literature. It is based on the assumption that complex traits or characters can only be gained once and lost once in the course of evolution. Here, we consider a generalization of this model, namely Dollo parsimony, that allows for multiple character losses. More precisely, we take a combinatorial

We introduce and study randomized sequential importance sampling algorithms for estimating the number of perfect matchings in bipartite graphs. In analyzing their performance, we establish various non-standard central limit theorems. We expect our methods to be useful for other applied problems.

The steady-state degree of a chemical reaction network is the number of complex steady-states, which is a measure of the algebraic complexity of solving the steady-state system. In general, the steady-state degree may be difficult to compute. Here, we give an upper bound to the steady-state degree of a reaction network by utilizing the underlying polyhedral geometry associated with the corresponding

We show that many well-known transforms in convex geometry (in particular, centroid body, convex floating body, and Ulam floating body) are special instances of a general construction, relying on applying sublinear expectations to random vectors in Euclidean space. We identify the dual representation of such convex bodies and describe a construction that serves as a building block for all so defined

Frieze patterns form a nexus between algebra, combinatorics, and geometry. T-paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables. This paper will introduce the concepts of weak friezes and T-paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call

Grinberg defined Nyldon words as those words which cannot be factorized into a sequence of lexicographically nondecreasing smaller Nyldon words. He was inspired by Lyndon words, defined the same way except with “nondecreasing” replaced by “nonincreasing.” Charlier, Philibert, and Stipulanti proved that, like Lyndon words, any word has a unique nondecreasing factorization into Nyldon words. They also

A matroid is uniform if and only if it has no minor isomorphic to U1,1⊕U0,1 and is paving if and only if it has no minor isomorphic to U2,2⊕U0,1. This paper considers, more generally, when a matroid M has no Uk,k⊕U0,ℓ-minor for a fixed pair of positive integers (k,ℓ). Calling such a matroid (k,ℓ)-uniform, it is shown that this is equivalent to the condition that every rank-(r(M)−k) flat of M has nullity

Let K be a finite, connected, abstract simplicial complex. The Morse complex of K, first introduced by Chari and Joswig, is the simplicial complex constructed from all gradient vector fields on K. We show that if K is neither the boundary of the n-simplex nor a cycle, then Aut(M(K))≅Aut(K). In the case where K=Cn, a cycle of length n, we show that Aut(M(Cn))≅Aut(C2n). When K=∂Δn, we prove that Aut

We develop a general framework for the probabilistic analysis of random finite point clouds in the context of topological data analysis. We extend the notion of a barcode of a finite point cloud to compact metric spaces. Such a barcode lives in the completion of the space of barcodes with respect to the bottleneck distance, which is quite natural from an analytic point of view. As an application we

Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties. We offer an alternative

In this paper, we consider the generalized phase retrieval from affine measurements. This problem aims to recover signals x∈Fd from the magnitude of the affine transformations yj=‖Mj⁎x+bj‖22,j=1,…,m, where Mj∈Fd×r,bj∈Fr,F∈{R,C} and we call it generalized affine phase retrieval. We first develop a framework for generalized affine phase retrieval with presenting several necessary and sufficient conditions

Let GR be the set of real points of a complex linear reductive group and Gˆλ, its classes of irreducible admissible representations with infinitesimal integral regular character λ. In this case, each cell of representations is associated to a special nilpotent orbit. This helps organize the corresponding set of irreducible Harish-Chandra modules. The goal of this paper is to bypass the need for the

In this paper we present some new identities for multiple polylogarithms (abbr. MPLs) and multiple harmonic star sums (abbr. MHSSs) by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish some explicit relations between Kaneko-Yamamoto type multiple zeta values (abbr. K-Y MZVs), multiple zeta values (abbr. MZVs) and MPLs

The notion of minimal complements was introduced by Nathanson in 2011. Since then, the existence or the inexistence of minimal complements of sets have been extensively studied. Recently, the study of inverse problems, i.e., which sets can or cannot occur as minimal complements has gained traction. For example, the works of Kwon, Alon–Kravitz–Larson, Burcroff–Luntzlara and also that of the authors

By combining the Gessel–Xin method with plethystic substitutions, we obtain a recursion for a symmetric function generalization of the q-Dyson constant term identity also known as the Zeilberger–Bressoud q-Dyson theorem. This yields a constant term identity which generalizes the non-zero part of Kadell's orthogonality ex-conjecture and a result of Károlyi, Lascoux and Warnaar.

Latin tableaux are a generalization of Latin squares, which first appeared in the early 2000's in a paper of Chow, Fan, Goemans, and Vondrák. Here, we extend the notion of isotopy, a permutation group action, from Latin squares to Latin tableaux. We define isotopy graphs for Latin tableaux, which encode the structure of orbits under the isotopy action, and investigate the relationship between the shape

The exponential generating function of the number Tn of standard Young tableaux of size n (or the number of involutions of n letters) is known to be et+t22, which coincides with the moment generating function of normal distribution N(1,1). We apply Laplace's method to the moment integral to derive a new asymptotic formula of Tn which is better than the known result. Meanwhile, by using of Can and Joyce's

The polygon valued Minkowski valuations in R2 intertwining with the special linear group are completely classified in this paper without any additional assumptions. Our results characterize projection bodies and difference bodies in R2. There are also some SL(2) intertwining Minkowski valuations that can only be defined in R2. It somehow suggests why such classification problems are more complicated

Many combinatorial numbers can be placed in the following generalized triangular array [Tn,k]n,k≥0 satisfying the recurrence relation:Tn,k=λ(a0n+a1k+a2)Tn−1,k+(b0n+b1k+b2)Tn−1,k−1+d(da1−b1)λ(n−k+1)Tn−1,k−2 with T0,0=1 and Tn,k=0 unless 0≤k≤n for suitable a0,a1,a2,b0,b1,b2,d and λ. For n≥0, denote by Tn(q) the generating function of the n-th row. In this paper, we develop various criteria for x-Stieltjes

We use the Mansour-Vainshtein theory of kernel shapes [14] to decompose the set Sn(1324) of 1324-avoiding permutations of length n into small pieces that are governed by some kernel shape λ. An enumeration of the set Km,c of all kernel shapes with length m and capacity c allows to express the generating function for the number of 1324-avoiding permutations of length n in terms of the Pm(C(x))=∑c|Km

We develop further the theory of q-deformations of real numbers introduced in [10] and [9] and focus in particular on the class of real quadratic irrationals. Our key tool is a q-deformation of matrices of the modular group PSL(2,Z). The action of the modular group by Möbius transformations commutes with the q-deformations. We prove that the traces of the q-deformed matrices are palindromic polynomials

Let Cn be the set of all permutation cycles of length n over {1,2,…,n}. Letfn(q):=∑σ∈Cn+1qmajσ be a q-analogue of the factorial n!, where maj denotes the major index. We prove a q-analogue of Wilson's congruencefn−1(q)≡μ(n)(modΦn(q)), where μ denotes the Möbius function and Φn(q) is the n-th cyclotomic polynomial.

The Markov numbers are the positive integer solutions of the Diophantine equation x2+y2+z2=3xyz. Already in 1880, Markov showed that all these solutions could be generated along a binary tree. So it became quite usual (and useful) to index the Markov numbers by the rationals from [0,1] which stand at the same place in the Stern–Brocot binary tree. The Frobenius' conjecture claims that each Markov number

We show that the sorting probability of the Catalan poset Pn satisfies δ(Pn)=O(n−5/4).

Non-orthogonal communication is a promising technique for future wireless networks (e.g., 6G and Wi-Fi 7). In the vector channel model, designing efficient non-orthogonal communication schemes amounts to the following extremum problem:max⁡mink⁡|vk|2σ2+∑l≠k〈vk,vl〉2 where the maximum is taken among vector systems (vk)1N⊂Rd satisfying c1⩽|vk|2⩽c2 for every k, and the parameter σ>0 corresponds to the noise

The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the surface areas of the involved sets, more precisely, by what is called the surface area deviation. The proof uses arguments and constructions from probability, convex and

Research on Helly-type theorems in combinatorial convex geometry has produced volumetric versions of Helly's theorem using witness sets and quantitative extensions of Doignon's theorem. This paper combines these philosophies and presents quantitative Helly-type theorems for the integer lattice with axis-parallel boxes as witness sets. Our main result shows that, while quantitative Helly numbers for

Phylogenetic networks are used to represent evolutionary relationships between species in biology. Such networks are often categorized into classes by their topological features, which stem from both biological and computational motivations. We study two network classes in this paper: tree-based networks and orchard networks. Tree-based networks are those that can be obtained by inserting edges between

A clique partition ε of graph G is a set of cliques such that each edge of G belongs to exactly one clique, and the total size of ε is the sum of cardinalities of all elements in ε. The ε-degree of a vertex u is the number of cliques in ε containing u. We say that ε is a k-restricted clique partition if each vertex has ε-degree at least k. The (k-restricted) clique partition number of G is the smallest

We introduce bijections between families of rooted maps with unfixed genus and families of so-called blossoming trees endowed with an arbitrary forward matching of their leaves. We first focus on Eulerian maps with controlled vertex degrees. The mapping from blossoming trees to maps is a generalization to unfixed genus of Schaeffer's closing construction for planar Eulerian maps. The inverse mapping

In this paper, we introduce the concept of weakly increasing trees on a multiset M, which is an extension of plane trees and increasing trees on the set {0,1,…,n}. We define the M-Eulerian–Narayana polynomial for weakly increasing trees on a multiset M, which interpolates between the Eulerian polynomial and the Narayana polynomial. We obtain a compact product formula for the number of weakly increasing

Inspired by problems in Private Information Retrieval, we consider the setting where two users need to establish a communication protocol to transmit a secret without revealing it to external observers. This is a question of how large a linear code can be, when it is required to agree with a prescribed code on a collection of coordinate sets. We show how the efficiency of such a protocol is determined

This paper aims to establish two bijections: from regions of the r-Shi arrangement to O-rooted labeled r-trees, and from regions of the r-Catalan arrangement to pairings of permutation and r-Dyck path. To this end, we introduce a cubic matrix for each region of the hyperplane arrangements. The first bijection is established by reading the positions of minimal positive entries in its column slices.

The notion of set-valued Young tableaux was introduced by Buch in his study of the Littlewood-Richardson rule for stable Grothendieck polynomials. Knutson, Miller and Yong showed that the double Grothendieck polynomials of 2143-avoiding permutations can be generated by flagged set-valued Young tableaux. In this paper, we introduce the structure of set-valued Rothe tableaux of permutations. Given the

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong σ-Gröbner bases. Moreover, we prove that the Terao's conjecture over finite fields implies the conjecture over the rationals.

We prove a general inclusion-exclusion relation for the extended chromatic symmetric function of a weighted graph, which specialises to (extended) k-deletion, and we give two methods to obtain numerous new bases from weighted graphs for the algebra of symmetric functions. Moreover, we classify when two weighted paths have equal extended chromatic symmetric functions by proving this is equivalent to

We study the enumeration problem for different kind of tree parking functions introduced recently, called tree parking functions, tree parking distributions, prime tree parking functions, and prime tree parking distributions, for rooted labelled trees of important combinatorial tree families including labelled ordered, unordered and binary trees. Using combinatorial decompositions of the underlying

Let Γ be a connected bridgeless metric graph, and fix a point v of Γ. We define combinatorial iterated integrals on Γ along closed paths at v, a unipotent generalization of the usual cycle pairing and the combinatorial analogue of Chen's iterated integrals on Riemann surfaces. These descend to a bilinear pairing between the group algebra of the fundamental group of Γ at v and the tensor algebra on

We introduce consecutive-pattern-avoiding stack-sorting maps SCσ, which are natural generalizations of West's stack-sorting map s and natural analogues of the classical-pattern-avoiding stack-sorting maps sσ recently introduced by Cerbai, Claesson, and Ferrari. We characterize the patterns σ such that Sort(SCσ), the set of permutations that are sortable via the map s∘SCσ, is a permutation class, and

The k-Stirling permutations are a generalization of both ordinary permutations and Stirling permutations introduced by Gessel and Stanley. In this paper, we introduce the statistics “linv” and “lmaj” on k-Stirling permutations and prove that they are equidistributed. This generalizes the famous result of MacMahon that the permutation statistics inversion number and major index are equidistributed.

We give a sufficient condition, in the case of 0

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

Abstract 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

Abstract Chvatal'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

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 class $\mathcal{E}$ of self-similar sets with overlaps. In particular, for a self-similar set $E\in \mathcal{E}$ and $k=1,2,\cdots, \aleph_0$ or $2^{\aleph_0}$ we investigate the Hausdorff dimension of the subset $U_k(E)$ which contains all points $x\in E$ having exactly $k$ different codings. This generalizes many results obtained in \cite{Dajani_Jiang_Kong_Li-2015} and

This paper provides a mathematical approach to study 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 spectrum

Abstract Szemeredi'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

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

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

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 \emph{reachable} if there is an IPF $(a,b)$ such that $x,y$ are