相关论文: On the enumeration of series-parallel matroids

We refine an idea of Deodhar, whose goal is a counting formula for Kazhdan-Lusztig polynomials. This is a consequence of a simple observation that one can use the solution of Soergel's conjecture to make ambiguities involved in defining…

We prove that the combinatorial concept of a special matching can be used to compute the parabolic Kazhdan-Lusztig polynomials of doubly laced Coxeter groups and of dihedral Coxeter groups. In particular, for this class of groups which…

We present a new, explicit sum formula for symmetric Macdonald polynomials $P_\lambda$ and show that they can be written as a trace over a product of (infinite dimensional) matrices. These matrices satisfy the Zamolodchikov--Faddeev (ZF)…

In 1999 Merino and Welsh conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article we show that the conjecture generalized to matroids holds for the large class of all split matroids by…

We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy…

We study the perfect matching lattice of a matching covered graph $G$, generated by the incidence vectors of its perfect matchings. Building on results of Lov\'asz and de Carvalho, Lucchesi, and Murty, we give a polynomial-time algorithm…

We design an FPRAS to count the number of bases of any matroid given by an independent set oracle, and to estimate the partition function of the random cluster model of any matroid in the regime where $0<q<1$. Consequently, we can sample…

In 2019, Perondi and Carmelo determined the set multipartite Ramsey number of particular complete bipartite graphs by establishing a relationship between the set multipartite Ramsey number, Hadamard matrices, and strongly regular graphs,…

In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove…

Panhandle matroids are a specific family of lattice-path matroids corresponding to panhandle-shaped Ferrers diagrams. Their matroid polytopes are the subpolytopes carved from a hypersimplex to form matroid polytopes of paving matroids. It…

The 1970s conjecture of Lov\'asz and Plummer that the number of perfect matchings in any $3$-regular graph is exponential in the number of vertices was proved in 2011 by Esperet, Kardo\v{s}, King, Kr\'al', and Norine. We give the exact…

A recent line of research has concentrated on exploring the links between analytic and combinatorial theories of submodularity, uncovering several key connections between them. In this context, Lov\'asz initiated the study of matroids from…

We describe various aspects of the Al-Salam-Chihara $q$-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization…

We prove that the extension complexity of the independence polytope of every regular matroid on $n$ elements is $O(n^6)$. Past results of Wong and Martin on extended formulations of the spanning tree polytope of a graph imply a $O(n^2)$…

The number of perfect matchings of a $k$-pfaffian graph can be counted by computing a linear combination of the pfaffians of $k$ matrices. The pfaffian number of a graph $G$ is the smallest integer $k$ such that $G$ is $k$-pfaffian. We…

In this note we introduce a family of polynomials on a matroid derived from chain Tutte polynomials which generalize the classic and ubiquitous characteristic polynomial. We show that the coefficients of these polynomials alternate and…

We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of…

Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements. By introducing augmented Chow rings of polymatroids, modeled after augmented…

In this work the interplay between matrix biorthogonal polynomials with respect to a matrix of linear functionals, the $k$-th associated matrix polynomials and the second kind matrix functions, is studied in terms of quasideterminants. A…

We use the Perron-Frobenius Theorem to define, study and, in some sense, classify special simple modules over arbitrary finite dimensional positively based algebras. For group algebras of finite Weyl groups with respect to the…