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

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.

We work out a graph limit theory for dense interval graphs. The theory developed departs from the usual description of a graph limit as a symmetric function W (x, y) on the unit square, with x and y uniform on the interval (0, 1). Instead, we fix a W and change the underlying distribution of the coordinates x and y. We find choices such that our limits are continuous. Connections to random interval

Recently, Sun (Two q-analogues of Euler’s formula \(\zeta (2)=\pi ^2/6\). arXiv:1802.01473, 2018) obtained q-analogues of Euler’s formula for \(\zeta (2)\) and \(\zeta (4)\). Sun’s formulas were based on identities satisfied by triangular numbers and properties of Euler’s q-Gamma function. In this paper, we obtain a q-analogue of \(\zeta (6)=\pi ^6/945\). Our main results are stated in Theorems 2.1

In this note, we give three identities for partitions with parts separated by parity, which were recently introduced by Andrews.