In his 1984 AMS memoir, Andrews introduced the family of k-colored generalized Frobenius partition functions. For any positive integer k, let cϕk(n) denote the number of k-colored generalized Frobenius partitions of n. Among many other things, Andrews proved that for any n≥0, cϕ2(5n+3)≡0(mod5). Since then, many scholars subsequently considered congruence properties of various k-colored generalized

We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue–Morse and Rudin–Shapiro substitutions, respectively. For such infinite words, we show that there exists a subsequence

The goal of the classic football-pool problem is to determine how many lottery tickets are to be bought in order to guarantee at least n−r correct guesses out of a sequence of n games played. We study a generalized (second-order) version of this problem, in which any of these n games consists of two sub-games. The second-order version of the football-pool problem is formulated using the notion of

We apply the methods of partition analysis to partitions with n copies of n. This allows us to obtain multivariable generating functions related to classical Rogers-Ramanujan type identities. Also, partitions with n copies of n are extended to partition diamonds yielding numerous new results including a natural connection to overpartitions and a variety of partition congruences.

List-decoding and list-recovery are important generalizations of unique decoding and receive considerable attention over the years. We study the optimal trade-off among the list-decoding (resp. list-recovery) radius, the list size, and the code rate, when the list size is constant and the alphabet size is large (both compared with the code length). We prove a new Singleton-type bound for list-decoding

Hypergraph Horn functions were introduced as a subclass of Horn functions that can be represented by a collection of circular implication rules. These functions possess distinguished structural and computational properties. In particular, their characterizations in terms of implicate-duality and the closure operator provide extensions of matroid duality and the Mac Lane – Steinitz exchange property

The universal enveloping algebra U(sl2) of sl2 is a unital associative algebra over C generated by E,F,H subject to the relations[H,E]=2E,[H,F]=−2F,[E,F]=H. The elementΛ=EF+FE+H22 is called the Casimir element of U(sl2). Let Δ:U(sl2)→U(sl2)⊗U(sl2) denote the comultiplication of U(sl2). The universal Hahn algebra H is a unital associative algebra over C generated by A,B,C and the relations assert that

We give a combinatorial proof of Stanley's shuffle theorem by using the insertion lemma of Haglund, Loehr and Remmel. Based on this combinatorial construction, we establish several refinements of Stanley's shuffle theorem.

Andrews recently provided a q-series proof of congruences for NT(m,k,n), the total number of parts in the partitions of n with rank congruent to m modulo k. Motivated by Andrews' works, Chern obtain congruences for Mω(m,k,n) which denotes the total number of ones in the partition of n with crank congruent to m modulo k. In this paper, we focus on the modular approach to these new partition statistics

The notion of multivariate P- and Q-polynomial association scheme has been introduced recently, generalizing the well-known univariate case. Numerous examples of such association schemes have already been exhibited. In particular, it has been demonstrated that the non-binary Johnson scheme is a bivariate P-polynomial association scheme. We show here that it is also a bivariate Q-polynomial association

We study representations of integral vectors in a number system with a matrix base M and vector digits. We focus on the case when M is equal or similar to Jn, the Jordan block with eigenvalue 1 and dimension n. If M=J2, we classify all digit sets of size two allowing representation for all of Z2. For M=Jn with n≥3, we show that a digit set of size three suffice to represent all of Zn. For bases M similar

Zagier provided eleven conjectural rank two examples for Nahm's problem. All of them have been proved in the literature except for the fifth example, and there is no q-series proof for the tenth example. We prove that the fifth and the tenth examples are in fact equivalent. Then we give a q-series proof for the fifth example, which confirms a recent conjecture of Wang. This also serves as the first

A family F of sets is union-closed if the union of any two sets from F belongs to F. The union-closed sets conjecture states that if F is a finite union-closed family of finite sets, then there is an element that belongs to at least half of the sets in F. The conjecture has several equivalent formulations in terms of other combinatorial structures such as lattices and graphs. In its whole generality

The problem of covering the ground set of two matroids by a minimum number of common independent sets is notoriously hard even in very restricted settings, i.e. when the goal is to decide if two common independent sets suffice or not. Nevertheless, as the problem generalizes several long-standing open questions, identifying tractable cases is of particular interest. Strongly base orderable matroids

A vertex triple (u,v,w) of a graph is called a 2-geodesic if v is adjacent to both u and w and u is not adjacent to w. A graph is said to be 2-geodesic transitive if its automorphism group is transitive on the set of 2-geodesics. In this paper, a complete classification of 2-geodesic transitive graphs of order pn is given for each prime p and n≤3. It turns out that all such graphs consist of three

A bijection which preserves five classical set-valued permutation statistics between (31245,32145,31254,32154)-avoiding permutations and (31425,32415,31524,32514)-avoiding permutations is constructed. Combining this bijection with two codings of permutations introduced respectively by Baril–Vajnovszki and Martinez–Savage, we prove an enumerative conjecture posed by Gao and Kitaev. Moreover, the generating

Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized

We show that the maximum cardinality of an equiangular line system in R18 is at most 59. Our proof includes a novel application of the Jacobi identity for complementary subgraphs. In particular, we show that there does not exist a graph whose adjacency matrix has characteristic polynomial (x−22)(x−2)42(x+6)15(x+8)2.

This paper introduces the notion of mesh patterns in multidimensional permutations and initiates a systematic study of singleton mesh patterns (SMPs), which are multidimensional mesh patterns of length 1. A pattern is avoidable if there exist arbitrarily large permutations that do not contain it. As our main result, we give a complete characterization of avoidable SMPs using an invariant of a pattern

We prove a new Elekes-Szabó type estimate on the size of the intersection of a Cartesian product A×B×C with an algebraic surface {f=0} over the reals. In particular, if A,B,C are sets of N real numbers and f is a trivariate polynomial, then either f has a special form that encodes additive group structure (for example, f(x,y,x)=x+y−z), or A×B×C∩{f=0} has cardinality O(N12/7). This is an improvement

In this paper, we formally introduce the concept of a row-sum matrix over an arbitrary group G. When G is cyclic, these types of matrices have been widely used to build uniform 2-factorizations of small Cayley graphs (or, Cayley subgraphs of blown-up cycles), which themselves factorize complete (equipartite) graphs. Here, we construct row-sum matrices over a class of non-abelian groups, the generalized

The Assmus-Mattson Theorem gives a way to identify block designs arising from codes. This result was broadened to matroids and weighted designs by Britz et al. in 2009. In this work we present a further two-fold generalisation: first from matroids to polymatroids and also from sets to vector spaces. To achieve this, we study the characteristic polynomial of a q-polymatroid and outline several of its

The Tutte polynomial is a well-studied invariant of matroids. The polymatroid Tutte polynomial TP(x,y), introduced by Bernardi, Kálmán, and Postnikov, is an extension of the classical Tutte polynomial from matroids to polymatroids P. In this paper, we first prove that TP(x,t) and TP(t,y) are interpolating for any fixed real number t≥1, and then we study the coefficients of high-order terms in TP(x

Let S be a non-empty finite set. A flag of S is a set f of non-empty proper subsets of S such that X⊆Y or Y⊆X for all X,Y∈f. The set {|X|:X∈f} is called the type of f. Two flags f and f′ are in general position with respect to S if X∩Y=∅ or X∪Y=S for all X∈f and Y∈f′. For a fixed type T, Klaus Metsch defined the general position graph Γ(S,T) whose vertices are the flags of S of type T with two vertices

Baxter permutations originally arose in studying common fixed points of two commuting continuous functions. In 2015, Dilks proposed a conjectured bijection between Baxter permutations and non-intersecting triples of lattice paths in terms of inverse descent bottoms, descent positions and inverse descent tops. We prove this bijectivity conjecture by investigating its connection with the Françon–Viennot

We classify an algebraic phenomenon on several families of wreath products that can be seen as coming from a generalization of a puzzle about switches on the corners of a spinning table. Such puzzles have been written about and generalized since they were first popularized by Martin Gardner in 1979. In this paper, we build upon a paper of Bar Yehuda, Etzion, and Moran, a paper of Ehrenborg and Skinner

The dimension of an irreducible representation of GL(n,C), Sp(2n), or SO(n) is given by the respective hook length and content formulas for the corresponding partition. The first author, inspired by the Nekrasov-Okounkov formula, conjectured combinatorial interpretations of analogous expressions involving hook lengths and symplectic/orthogonal contents. We prove special cases of these conjectures.

If S is a transitive metric space, then |C|⋅|A|≤|S| for any distance-d code C and a set A, “anticode”, of diameter less than d. For every Steiner S(t,k,n) system S, we show the existence of a q-ary constant-weight code C of length n, weight k (or n−k), and distance d=2k−t+1 (respectively, d=n−t+1) and an anticode A of diameter d−1 such that the pair (C,A) attains the code–anticode bound and the supports

This paper is concerned with five kinds of modification of hyperplane arrangements, including elementary lift, parallel translation, coning, one-element extension and restriction to a hyperplane. We show that the combinatorial classification of all hyperplane arrangements of each kind of modification will be characterized by the intersection lattice of the discriminantal or adjoint arrangement. Based

A linear configuration is said to be common in an Abelian group G if every 2-coloring of G yields at least the number of monochromatic instances of a randomly chosen coloring. Saad and Wolf asked whether, analogously to a result by Jagger, Šťovíček and Thomason in graph theory, every configuration containing a 4-term arithmetic progression is uncommon. We prove this in Fpn for p≥5 and large n and in

We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by powers of (1−ζ), where ζ is a root of unity. We also prove a generalisation of a classical result of Harary and Schwenk about a relation for traces of powers of

In this paper, we study sum formulas for Schur multiple zeta values and give a generalization of the sum formulas for multiple zeta(-star) values. We show that for ribbons of certain types, the sum of Schur multiple zeta values over all admissible Young tableaux of this shape evaluates to a rational multiple of the Riemann zeta value. For arbitrary ribbons with n corners, we show that such a sum can

Two families F,G of k-subsets of {1,2,…,n} are called non-trivial cross-intersecting if F∩G≠∅ for all F∈F,G∈G and ∩{F:F∈F}=∅=∩{G:G∈G}. In the present paper, we determine the maximum product of the sizes of two non-trivial cross-intersecting families of k-subsets of {1,2,…,n} for n≥4k, k≥8, which is a product version of the classical Hilton-Milner Theorem.

Recently, 4-regular partitions into distinct parts were connected with a family of overpartitions. In this paper, we provide a uniform extension of two relations due to Andrews for the two types of partitions. Such an extension is made possible with recourse to a new trivariate Rogers–Ramanujan type identity, which concerns a family of quadruple summations appearing as generating functions for the

The research on orthogonal Steiner systems S(t,k,v) was initiated in 1968. For (t,k)∈{(2,3),(3,4)}, this corresponds to orthogonal Steiner triple systems (STSs) and Steiner quadruple systems (SQSs), respectively. The existence problem of a pair of orthogonal STSs or SQSs was settled completely thirty years ago. However, for Steiner systems with t≥3 and k≥5, only two small examples of orthogonal pairs

In 1983, Burr and Erdős initiated the study of Ramsey goodness problems. Nikiforov and Rousseau (2009) resolved almost all goodness questions raised by Burr and Erdős, in which the bounds on the parameters are of tower type since their proofs rely on the regularity lemma. Let Bk,n be the book graph on n vertices which consists of n−k copies of Kk+1 all sharing a common Kk, and let H=Kp(a1,…,ap) be

A graph G=(V,E) is called fully regular if for every independent set I⊂V, the number of vertices in V∖I that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G is called successive if for every i, the first i vertices induce a connected subgraph of G. We give an explicit formula for the number of successive vertex orderings of a fully regular

Recently Lazar and Wachs proved two new permutation models, called D-permutations and E-permutations, for Genocchi and median Genocchi numbers. In a follow-up, Eu et al. studied the even-odd descent permutations, which are in bijection with E-permutations. We generalize Eu et al.'s descent polynomials with eight statistics and obtain an explicit J-fraction formula for their ordinary generaing function

The development of the theory of the second-order Eulerian polynomials began with the works of Buckholtz and Carlitz in their studies of an asymptotic expansion. Gessel-Stanley introduced Stirling permutations and provided combinatorial interpretations for the second-order Eulerian polynomials in terms of Stirling permutations. The Stirling permutations have been extensively studied by many researchers

Constructing bent functions by composing a Boolean function with a permutation was introduced by Hou and Langevin in 1997. The approach appears simple but heavily depends on the construction of desirable permutations. In this paper, we further study this approach by investigating the exponential sums of certain monomials and permutations. We propose several classes of bent functions from quadratic

For a set of positive integers A⊆[n], an r-coloring of A is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erdős-Rothschild problem in the context of sum-free sets, which asks for the subsets of [n] with the maximum number of rainbow sum-free r-colorings. We show that for r=3, the interval [n] is optimal, while for r≥8, the set [⌊n/2⌋,n]

Abstract not available

It is known that isomorphisms of graph Jacobians induce cyclic bijections on the associated graphs. We characterize when such cyclic bijections can be strengthened to graph isomorphisms, in terms of an easily computed divisor. The result refines tools used in algebraic geometry to examine the fibers of the compactified Torelli map.

The “Kruskal-Katona-type problem for a graph G” concerned here is to describe subsets of vertices of G that have minimum number of neighborhoods with respect to their sizes. In this paper, we establish a Kruskal-Katona-type theorem for the q-Kneser graphs, whose vertex set consists of all k-dimensional subspaces of an n-dimensional linear space over a q-element field, two subspaces are adjacent if

Recently, regular Cayley maps of cyclic groups and dihedral groups have been classified in [7] and [20], respectively. A nature question is to classify regular Cayley maps of elementary abelian p-groups Zpn. In this paper, a complete classification of regular Cayley maps of Zpn is given and moreover, the number of these maps and their genera are enumerated.

The famous Erdős-Falconer distance problems aim to quantify the extent to which large sets must determine many distinct distances. The problems in both the Euclidean setting and the discrete setting have received much attention. An interesting generalization of Erdős-Falconer problems is to find locating point configurations with prescribed metric structure within large subsets. In this paper, we consider

Multi-Schur functions are symmetric functions that generalize the supersymmetric Schur functions, the flagged Schur functions, and the refined dual Grothendieck functions, which have been intensively studied by Lascoux. In this paper, we give a new free-fermionic presentation of them. The multi-Schur functions are indexed by a partition and two “tuples of tuples” of indeterminates. We construct a family

New nonexistence results of generalized bent functions from Z2n to Zm are presented. In particular, we prove that there exists no generalized bent function from Z24 to Zm for every odd m.

Given a square matrix A over the integers, we consider the Z-module MA generated by the set of all matrices that are permutation-similar to A. Motivated by analogous problems on signed graph decompositions and block designs, we are interested in the completely symmetric matrices aI+bJ belonging to MA. We give a relatively fast method to compute a generator for such matrices, avoiding the need for a

The interior and exterior activities of bases of a matroid are well-known notions that for instance permit one to define the Tutte polynomial. Recently, we have discovered correspondences between the regions of gainic hyperplane arrangements and colored labeled rooted trees. Here we define a general activity theory that applies in particular to no-broken circuit (NBC) sets and labeled colored trees

Abstract not available

We give a complete characterization of all indecomposable involutive solutions of the Yang-Baxter equation of multipermutation level 2. In the first step we present a construction of some family of such solutions and in the second step we prove that every indecomposable involutive solution of the Yang-Baxter equation with multipermutation level 2 is a homomorphic image of a solution previously constructed

A map is called a p-map if it has a prime p-power vertices. An orientably-regular (resp. A regular) p-map is called solvable if the group G+ of all orientation-preserving automorphisms (resp. the group G of automorphisms) is solvable; and called normal if G+ (resp. G) contains the normal Sylow p-subgroup. In this paper, it will be proved that both orientably-regular p-maps and regular p-maps are solvable

We present (bi-)symmetric generating functions for the joint distributions of Euler–Stirling statistics on permutations, including the number of descents (des), inverse descents (ides), the number of left-to-right maxima (lmax), the number of right-to-left maxima (rmax) and the number of left-to-right minima (lmin). We also show how they recover the classical symmetric generating function for permutations

The pairs (D,G), where D is a non-trivial 2-(k2,k,λ) design with λ|k and G is a flag-transitive automorphism group of D are classified except for k a power of a prime and G⩽AΓL1(k2).

Let Cn be the cyclic group of order n. In this paper, we provide the exact values of some zero-sum constants over G=Cn⋊sC2 where s≢±1(modn), namely η-constant, Gao constant, and Erdős-Ginzburg-Ziv constant (the latter for all but a “small” family of cases). As a consequence, we prove the Gao's and Zhuang-Gao's Conjectures for groups of this form. We also solve the associated inverse problems by characterizing

Dorpalen-Barry et al. proved Elser's conjecture about signs of Elser's numbers by interpreting them as certain sums of reduced Euler characteristic of an abstract simplicial complex known as U-nucleus complex. We prove a conjecture posed by them regarding the homology of U-nucleus complex.

We prove that the number of even parts and the number of times that parts are repeated have the same distribution over integer partitions with a fixed perimeter. This refines Straub's analog of Euler's Odd-Distinct partition theorem. We generalize the two concerned statistics to those of the part-difference less than d and the parts not congruent to 1 modulo d+1 and prove a distribution inequality

To a simple graph we associate a so-called graph series, which can be viewed as the Hilbert–Poincaré series of a certain infinite jet scheme. We study new q-representations and examine modular properties of several examples including Dynkin diagrams of finite and affine type. Notably, we obtain new formulas for graph series of type A7 and A8 in terms of “sum of tails” series, and of type D4 and D5

A conjectured relation between Ramanujan's asymptotic approximations to the exponential function and the exponential integral is established. The proof involves Stirling numbers, second-order Eulerian numbers, modifications of both of these, and Stirling's approximation to the gamma function. Our work provides new information about the coefficients in Stirling's approximation and their connection to