The main idea of this paper is to provide an algebraic algorithm for constructing symmetric graphs with optimal fault tolerance. For this purpose, we use normal edge-transitive Cayley graphs and the idea of reconstruction question posed by Praeger to present a special factorization of groups which induces a graphical decomposition of normal edge-transitive Cayley graphs to simpler normal edge-transitive

Lock polynomials and lock tableaux are natural analogues to key polynomials and Kohnert tableaux, respectively. In this paper, we compare lock polynomials to the much-studied key polynomials and give an explicit description of a crystal structure on lock tableaux. Furthermore, we construct an injective, weight-preserving map from lock tableaux to Kohnert tableaux that intertwines with their respective

The q, t-Catalan number \({{\,\mathrm{Cat}\,}}_n(q,t)\) enumerates integer partitions contained in an \(n\times n\) triangle by their dinv and external area statistics. The paper by Lee et al. (SIAM J Discr Math 32:191–232, 2018) proposed a new approach to understanding the symmetry property \({{\,\mathrm{Cat}\,}}_n(q,t)={{\,\mathrm{Cat}\,}}_n(t,q)\) based on decomposing the set of all integer partitions

A power series is called lacunary if “almost all” of its coefficients are zero. Integer partitions have motivated the classification of lacunary specializations of Han’s extension of the Nekrasov–Okounkov formula. More precisely, we consider the modular forms $$\begin{aligned}F_{a,b,c}(z) :=\frac{\eta (24az)^a \eta (24acz)^{b-a}}{\eta (24z)},\end{aligned}$$ defined in terms of the Dedekind \(\eta \)-function

The Eulerian polynomial \( \mathrm {AExc}_n(t)\) enumerating excedances in the symmetric group \(\mathfrak {S}_n\) is known to be gamma positive for all n. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also gamma positive for all n. We consider \( \mathrm {AExc}_n^+(t)\) and \( \mathrm {AExc}_n^-(t)\), the polynomials which enumerate

The regularity \({\text {reg}}R(I(G))\) of the Rees ring R(I(G)) of the edge ideal I(G) of a finite simple graph G is studied. It is shown that, if R(I(G)) is normal, one has \({\text {mat}}(G) \le {\text {reg}}R(I(G)) \le {\text {mat}}(G) + 1\), where \({\text {mat}}(G)\) is the matching number of G. In general, the induced matching number is a lower bound for the regularity, which can be shown by

For a given marked surface (S, M) and a fixed tagged triangulation T of (S, M), we show that each tagged triangulation \(T'\) of (S, M) is uniquely determined by the intersection numbers of tagged arcs of T and tagged arcs of \(T'\). As a consequence, each cluster in the cluster algebra \({{\,\mathrm{{\mathcal {A}}}\,}}(T)\) is uniquely determined by its F-matrix which is a new numerical invariant

We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an R-labeling, imply the existence of the (non-homogeneous) \(\mathbf{c}\mathbf{d}\)-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality

In this paper, we investigate a generalization of the Bessenrodt–Ono inequality by following Gian–Carlo Rota’s advice in studying problems in combinatorics and number theory in terms of roots of polynomials. We consider the number of k-colored partitions of n as special values of polynomials \(P_n(x)\). We prove for all real numbers \(x >2 \) and \(a,b \in \mathbb {N}\) with \(a+b >2\) the inequality:

Polymatroids can be considered as “fractional matroids” where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to polymatroids. Defining cyclic flats of a polymatroid carefully, the characterization by Bonin and de Mier of the ranked lattice of cyclic flats carries over to polymatroids. The main tool, which might be of independent

We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then projecting the coefficients to QSym. By duality, we obtain a basis of noncommutative symmetric functions, for which a product formula and a recurrence in the form of a combinatorial

The Lucas sequence is a sequence of polynomials in s, t defined recursively by \(\{0\}=0\), \(\{1\}=1\), and \(\{n\}=s\{n-1\}+t\{n-2\}\) for \(n\ge 2\). On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the q-integers \([n]_q\). Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue

By an oval in \({\mathbb {Z}}^2_{2p},\)p odd prime, we mean a set of \(2p+2\) points, such that no three of them are on a line. It is shown that ovals in \({\mathbb {Z}}^2_{2p}\) only exist for \(p=3,5\) and they are unique up to an isomorphism.

An inversion triple of an element w of a simply laced Coxeter group W is a set \(\{ \alpha , \beta , \alpha + \beta \}\), where each element is a positive root sent negative by w. We say that an inversion triple of w is contractible if there is a root sequence for w in which the roots of the triple appear consecutively. Such triples arise in the study of the commutation classes of reduced expressions

Recently it was shown that, if the subtree and chain reduction rules have been applied exhaustively to two unrooted phylogenetic trees, the reduced trees will have at most \(15k-9\) taxa where k is the TBR (Tree Bisection and Reconnection) distance between the two trees, and that this bound is tight. Here, we propose five new reduction rules and show that these further reduce the bound to \(11k-9\)

Kostant’s partition function is a vector partition function that counts the number of ways one can express a weight of a Lie algebra \(\mathfrak {g}\) as a nonnegative integral linear combination of the positive roots of \(\mathfrak {g}\). Multiplex juggling sequences are generalizations of juggling sequences that specify an initial and terminal configuration of balls and allow for multiple balls at

Weights of permutations were originally introduced by Dugan et al. (Journal of Combinatorial Theory, Series A 164:24–49, 2019) in their study of the combinatorics of tiered trees. Given a permutation \(\sigma \) viewed as a sequence of integers, computing the weight of \(\sigma \) involves recursively counting descents of certain subpermutations of \(\sigma \). Using this weight function, one can define

Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are Cayley graphs of abelian groups. Such groups can be constructed using a generalization to \(\mathbb {Z}^n\) of the concept of congruence in \(\mathbb {Z}\). Here we use this approach to present some families of mixed graphs, which, for

We present new applications of q-binomials, also known as Gaussian binomial coefficients. Our main theorems determine cardinalities of certain error-correcting codes based on Varshamov–Tenengolts codes and prove a curious phenomenon relating to deletion spheres.

Ramanujan’s modular equations of degrees 3, 5, 7, 11 and 23 yield beautiful colored partition identities. Warnaar analytically generalized the modular equations of degrees 3 and 7, and thereafter, the author found bijective proofs of those partitions identities and recently, established an analytic generalization of the modular equations of degrees 5, 11 and 23. The partition identities of degrees

A matroid is Gorenstein if its toric variety is. Hibi, Lasoń, Matsuda, Michałek, and Vodička provided a full graph-theoretic classification of Gorenstein matroids associated to simple graphs. We extend this classification to multigraphs.

Let \({{\mathfrak {S}}}_n\) be the nth symmetric group. Given a set of permutations \(\Pi \), we denote by \({{\mathfrak {S}}}_n(\Pi )\) the set of permutations in \({{\mathfrak {S}}}_n\) which avoid \(\Pi \) in the sense of pattern avoidance. Consider the generating function \(Q_n(\Pi )=\sum _\sigma F_{{{\,\mathrm{Des}\,}}\sigma }\) where the sum is over all \(\sigma \in {{\mathfrak {S}}}_n(\Pi )\)

The Deodhar decomposition of the Grassmannian is a refinement of the Schubert, Richardson, and positroid stratifications of the Grassmannian. Go-diagrams are certain fillings of Ferrers diagrams with black stones, white stones, and pluses which index Deodhar components in the Grassmannian. We provide a series of corrective flips on diagrams which may be used to transform arbitrary fillings of Ferrers

In this paper, we utilize the machinery of cluster algebras, quiver mutations, and brane tilings to study a variety of historical enumerative combinatorics questions all under one roof. Previous work (Zhang in Cluster variables and perfect matchings of subgraphs of the \(dP_{3}\) lattice, http://www.math.umn.edu/~reiner/REU/Zhang2012.pdf. arXiv:1511.0655, 2012; Leoni et al. in J Phys A Math Theor 47:474011

The Murnaghan–Nakayama rule is a combinatorial rule for the character values of symmetric groups. We give a new combinatorial proof by explicitly finding the trace of the representing matrices in the standard basis of Specht modules. This gives an essentially bijective proof of the rule. A key lemma is an extension of a straightening result proved by the second author to skew tableaux. Our module theoretic

For a given w in a Coxeter group W, the elements u smaller than w in Bruhat order can be seen as the end alcoves of stammering galleries of type w in the Coxeter complex \(\Sigma \). We generalize this notion and consider sets of end alcoves of galleries that are positively folded with respect to certain orientation \(\phi \) of \(\Sigma \). We call these sets shadows. Positively folded galleries are

In a series of papers, Dartyge and Sárközy (partly with other coauthors) studied pseudorandom subsets. In this paper, we study the minimal values of correlation measures of pseudorandom subsets by extending the methods introduced in Alon et al. (Comb Probab Comput 15:1–29, 2006), Anantharam (Discrete Math 308:6203–6209, 2008), Gyarmati (Stud Sci Math Hung 42:79–93, 2005) and Gyarmati and Mauduit (Discrete

The degree of a lattice polytope is a notion in Ehrhart theory that was studied quite intensively over previous years. It is well known that a lattice polytope has normalized volume one if and only if its degree is zero. Recently, Esterov and Gusev gave a complete classification result for families of n lattice polytopes in \({\mathbb {R}}^n\) whose mixed volume equals one. Here, we give a reformulation

We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength n that is shown to be in bijection with the set of all SYT with n boxes. In addition, we present bijections

For a poset whose Hasse diagram is a rooted plane forest F, we consider the corresponding tree descent polynomial \(A_F(q)\), which is a generating function of the number of descents of the labelings of F. When the forest is a path, \(A_F(q)\) specializes to the classical Eulerian polynomial. We prove that the coefficient sequence of \(A_F(q)\) is unimodal and that if \(\{T_{n}\}\) is a sequence of

A famous theorem of Euler asserts that there are as many partitions of n into distinct parts as there are partitions into odd parts. The even parts in partitions of n into distinct parts play an important role in the Euler–Glaisher bijective proof of this result. In this paper, we investigate the number of even parts in all partitions of n into distinct parts providing new combinatorial interpretations

In this article, we prove a tableau formula for the double Grothendieck polynomials associated to 321-avoiding permutations. The proof is based on the compatibility of the formula with the K-theoretic divided difference operators. Our formula specializes to the one obtained by Chen et al. (Eur J Combin 25(8):1181–1196, 2004) for the (double) skew Schur polynomials.

We define operators on semistandard shifted tableaux and use Stembridge’s local characterization for regular graphs to prove they define a crystal structure. This gives a new proof that Schur P-polynomials are Schur positive. We define queer crystal operators (also called odd Kashiwara operators) to construct a connected queer crystal on semistandard shifted tableaux of a given shape. Using the tensor

In work with A. Yong, the author introduced genomic tableaux to prove the first positive combinatorial rule for the Littlewood–Richardson coefficients in torus-equivariant K-theory of Grassmannians. We then studied the genomic Schur function \(U_\lambda \), a generating function for such tableaux, showing that it is nontrivially a symmetric function, although generally not Schur-positive. Here, we

Let \({\mathcal {P}}\) be a set of n points in the Euclidean plane. We prove that, for any \(\varepsilon > 0\), either a single line or circle contains n/2 points of \({\mathcal {P}}\), or the number of distinct perpendicular bisectors determined by pairs of points in \({\mathcal {P}}\) is \(\Omega (n^{52/35 - \varepsilon })\), where the constant implied by the \(\Omega \) notation depends on \(\varepsilon

The monopole-dimer model introduced recently is an exactly solvable signed generalisation of the dimer model. We show that the partition function of the monopole-dimer model on a graph invariant under a fixed-point free involution is a perfect square. We give a combinatorial interpretation of the square root of the partition function for such graphs in terms of a monopole-dimer model on a new kind

Complete partitions are a generalization of MacMahon’s perfect partitions; we further generalize these by defining k-step partitions. A matrix equation shows an unexpected connection between k-step partitions and distinct part partitions. We provide two proofs of the corresponding theorem, one using generating functions and one combinatorial. The algebraic proof relies on a generalization of a conjecture

The class of permutations that avoid the bivincular pattern \((231, \{1\},\{1\})\) is known to be enumerated by the Fishburn numbers. In this paper, we call them Fishburn permutations and study their pattern avoidance. For classical patterns of size 3, we give a complete enumerative picture for regular and indecomposable Fishburn permutations. For patterns of size 4, we focus on a Wilf equivalence

Let \(L \in \{6,9,12 \} \). We determine the generating functions of certain combinations of three ranks and three cranks modulo L in terms of eta quotients. Then, using the periodicity of signs of these eta quotients, we compare their values with the values of \(\frac{p(n)}{L/3}\).

If \(p_D(n)\) denotes the number of partitions of n into distinct parts, it is known that for \(\alpha \ge 1\) and \(n\ge 0\),$$\begin{aligned} p_D\left( 5^{2\alpha +1}n+\frac{5^{2\alpha +2}-1}{24}\right) \equiv 0\pmod {5^\alpha }. \end{aligned}$$We give a completely elementary proof of this fact.

The goal of this paper is to introduce and study noncommutative Catalan numbers\(C_n\) which belong to the free Laurent polynomial algebra \(\mathcal {L}_n\) in n generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia–Haiman (q, t)-versions, another—to solving noncommutative quadratic equations. We also establish total

We construct Andrews–Gordon type positive series as generating functions of partitions satisfying certain difference conditions in six conjectures by Kanade and Russell. Thus, we obtain q-series conjectures as companions to Kanade and Russell’s combinatorial conjectures. We construct generating functions for missing partition enumerants as well, without claiming new partition identities.

In this paper, we place the work of Andrews et al. (Invent Math 91(3):391–407, 1988) and Cohen (Invent Math 91(3):409–422, 1988), relating arithmetic in \({{\mathbb {Q}}}(\sqrt{6})\) to modularity of Ramanujan’s function \(\sigma (q)\), in the context of the general family of Richaud–Degert real quadratic fields \({{\mathbb {Q}}}(\sqrt{2p})\). Moreover, we give the resulting generalizations of the

This paper is concerned with a class of partition functions a(n) introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radu’s algorithms, we present an algorithm to find Ramanujan-type identities for \(a(mn+t)\). While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness

We report on findings of a variant of IdentityFinder—a Maple program that was used by two of the authors to conjecture several new identities of Rogers–Ramanujan kind. In the present search, we modify the parametrization of the search space by taking into consideration several aspects of Lepowsky and Wilson’s Z-algebraic mechanism and its variant by Meurman and Primc. We search for identities based

Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial interpretation in terms of choosing subsets of sets with a negative number of elements. We show that all of this can be extended to the case of Gaussian binomial coefficients

We introduce a polynomial E(d, t, x) in three variables that comes from the intersections of a family of ellipses described by Euler. For fixed odd integers \(t\ge 3\), the sequence of E(d, t, x) with d running through the integers produces, conjecturally, sequences of “twin composites” analogous to the twin primes of the integers. This polynomial and its lower degree relative R(d, t, x) have strikingly

We establish an algorithm for producing formulas for p(n, m, N), the function enumerating partitions of n into at most m parts with no part larger than N. Recent combinatorial results of H. Hahn et al. on a collection of partition identities for p(n, 3, N) are considered. We offer direct proofs of these identities and then place them in a larger context of the unimodality of Gaussian polynomials \(N+m\brack

Given an integer base \(b\ge 2\), we investigate a multivariate b-ary polynomial analogue of Stern’s diatomic sequence which arose in the study of hyper b-ary representations of integers. We derive various properties of these polynomials, including a generating function and identities that lead to factorizations of the polynomials. We use some of these results to extend an identity of Courtright and

We consider certain classes of compositions of numbers based on the recently introduced extension of conjugation to higher orders. We use generating functions and combinatorial identities to provide enumeration results for compositions possessing conjugates of a given order. Working under some popular themes in the theory, we show that results for these compositions specialize to standard results in

We present an infinite family of Borwein type \(+ - - \) conjectures. The expressions in the conjecture are related to multiple basic hypergeometric series with Macdonald polynomial argument.

Singular overpartitions, which are Frobenius symbols with at most one overlined entry in each row, were first introduced by Andrews in 2015. In his paper, Andrews investigated an interesting subclass of singular overpartitions, namely, (K, i)-singular overpartitions for integers K, i with \( 1\le i

We discuss q-analogues of the classical congruence \(\left( {\begin{array}{c}ap\\ bp\end{array}}\right) \equiv \left( {\begin{array}{c}a\\ b\end{array}}\right) \pmod {p^3}\), for primes \(p>3\), as well as its generalisations. In particular, we prove related congruences for (q-analogues of) integral factorial ratios.

We give a simple and a more explicit proof of a mod 4 congruence for a series involving the little q-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function.

In this paper, we generalize the Andrews–Yee identities associated with the third-order mock theta functions \(\omega (q)\) and \(\nu (q)\). We obtain some q-series transformation formulas, one of which gives a new Bailey pair. Using the classical Bailey lemma, we derive a product formula for two \({}_2\phi _1\) series. We also establish recurrence relations and transformation formulas for two finite

Usually, the Weierstraß gap theorem is derived as a straightforward corollary of the Riemann–Roch theorem. Our main objective in this article is to prove the Weierstraß gap theorem by following an alternative approach based on “first principles”, which does not use the Riemann–Roch formula. Having mostly applications in connection with modular functions in mind, we describe our approach for the case

MacMahon showed that the generating function for partitions into at most k parts can be decomposed into a partial fraction-type sum indexed by the partitions of k. In the present work, a generalization of MacMahon’s result is given, which in turn provides a full combinatorial explanation.

In his lost notebook, Ramanujan listed five identities related to the false theta function:$$\begin{aligned} f(q)=\sum _{n=0}^\infty (-1)^nq^{n(n+1)/2}. \end{aligned}$$A new combinatorial interpretation and a proof of one of these identities are given. The methods of the proof allow for new multivariate generalizations of this identity. Additionally, the same technique can be used to obtain a combinatorial

We give simple proofs of Hecke–Rogers indefinite binary theta series identities for the two Ramanujan’s fifth order mock theta functions \(\chi _0(q)\) and \(\chi _1(q)\) and all three of Ramanujan’s seventh order mock theta functions. We find that the coefficients of the three mock theta functions of order 7 are surprisingly related.

In this note, we show how to use cylindric partitions to rederive the four \(A_2\) Rogers–Ramanujan identities originally proven by Andrews, Schilling and Warnaar, and provide a proof of a similar fifth identity.