Related papers: On distinguishing digraphs by its quasisymmetric B…
We demonstrate that a system of bi-orthogonal polynomials and their associated functions corresponding to a regular semi-classical weight on the unit circle constitute a class of general classical solutions to the Garnier systems by…
The derangement polynomial for the symmetric group enumerates derangements by the number of excedances. It can be interpreted as the local $h$-polynomial, in the sense of Stanley, of the barycentric subdivision of the simplex. Motivated by…
Partially ordered patterns (POPs) generalize the classical notion of permutation patterns within the framework of pattern avoidance. Building on recent work by Burstein, Han, Kitaev, and Zhang, which introduced the concept of…
We construct a complex of toric varieties we call the quasisymmetric Grassmannian inside the Grassmannian of $r$-planes in $\mathbb{C}^n$. Each irreducible component is a positroid variety and an $S_n$ translate of a toric Richardson…
Grothendieck's theory of dessins provides a bridge between algebraic numbers and combinatorics. This paper adds a new concept, called 'bias', to the bridge. This produces: (i) from a biased plane tree the construction of a sequence of…
Chisini's conjecture asserts that for a cuspidal curve $B\subset \mathbb P^2$ a generic morphism $f$ of a smooth projective surface onto $\mathbb P^2$ of degree $\geq 5$, branched along $B$, is unique up to isomorphism. We prove that if…
A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. Fixing a total order on the edges and a reference orientation of the graph, we…
The symmetric Macdonald polynomials are able to be constructed out of the non-symmetric Macdonald polynomials. This allows us to develop the theory of the symmetric Macdonald polynomials by first developing the theory of their non-symmetric…
In this paper we introduce and investigate a one-parameter family of polynomials. They are semisymmetric, i.e. symmetric in the variables with odd and even index separately. In fact, the family forms a basis of the space of semisymmetric…
We study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices with respect to their inversion number, complementary inversion number and the position of the…
We show that a family of multivariate polynomials recently introduced by Bessenrodt and Stanley can be expressed as solution of the octahedron recurrence with suitable initial data. This leads to generalizations and explicit expressions as…
The hyperoctahedral group is the Weyl group of type B and is associated with a two-parameter family of differential-difference operators T_i, i=1,..,N (the dimension of the underlying Euclidean space). These operators are analogous to…
Let $G$ be a finite simple graph and $B \subseteq V(G)$. We introduce the \emph{bivariate $B$-restricted clique polynomial} \[ C_B(G;x,y) = \sum_{\substack{K \subseteq V \\ K \text{ is a clique}}} x^{|K|} y^{|K \cap B|}, \] where the…
A tight map is a map with some of its vertices marked, such that every vertex of degree $1$ is marked. We give an explicit formula for the number $N_{0,n}(d_1,\ldots,d_n)$ of planar tight maps with $n$ labeled faces of prescribed degrees…
We study the PBW filtration on irreducible finite--dimensional representations for the Lie algebra of type $\tt B_n$. We prove in several cases, including all multiples of the adjoint representation and all irreducible finite--dimensional…
Let $m\ge 1$ be an integer and $G$ be a graph with $m$ edges. We say that $G$ has an antimagic orientation if $G$ has an orientation $D$ and a bijection $\tau:A(D)\rightarrow \{1,2,\ldots,m\}$ such that no two vertices in $D$ have the same…
The deep interconnection between linear algebra and graph theory allows one to interpret classical matrix invariants through combinatorial structures. To each square matrix A over a commutative ring K, one can associate a weighted directed…
We consider, for complete bipartite graphs, the convex hulls of characteristic vectors of all matchings, extended by a binary entry indicating whether the matching contains two specific edges. These polytopes are associated to the quadratic…
With every family of finitely many subsets of a finite-dimensional vector space over the Galois-field with two elements we associate a cyclic transversal polytope. It turns out that those polytopes generalize several well-known polytopes…
Superpolynomials consist of commuting and anti-commuting variables. By considering the anti-commuting variables as a module of the symmetric group the theory of vector-valued nonsymmetric Jack polynomials can be specialized to…