Related papers: Lattice path matroids and quotients

We conduct a systematic study of the Ehrhart theory of certain slices of rectangular prisms. Our polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov;…

There exist several theorems which state that when a matroid is representable over distinct fields F_1,...,F_k, it is also representable over other fields. We prove a theorem, the Lift Theorem, that implies many of these results. First,…

Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers such questions as the enumeration of the number of nonzero coefficients of $f$ or of the number of coefficients equal to $\alpha\in K^*$. For instance, if $K=\ff_q$…

We study the partial orders induced on Wachs and signed Wachs permutations by the Bruhat and weak orders of the symmetric and hyperoctahedral groups. We show that these orders are graded, determine their rank function, characterize their…

We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence $(1, 4, 4^2, 4^3, ...)$ which has been derived by Shapiro, Woan and Getu by using Riordan arrays. By giving a bijection between weighted partial…

The immanants of combinatorial matrices have many significant properties, including m-positivity and Schur positivity. While the immanants of Jacobi-Trudi matrices are known to be both m-positive and Schur positive, those of Giambelli…

It is already known that order polytopes and chain polytopes are always 2-level polytopes. In general, this is not true for marked order and marked chain polytopes. We study the geometry of marked order polytopes, marked chain polytopes,…

For any marked poset we define a continuous family of polytopes, parametrized by a hypercube, generalizing the notions of marked order and marked chain polytopes. By providing transfer maps, we show that the vertices of the hypercube…

We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP.…

Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if…

We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third…

This article is a survey of matroid theory aimed at algebraic geometers. Matroids are combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all matroids come from linear subspaces; those that do are said to be…

Gao and Xie (2021) conjectured that the inverse Kazhdan-Lusztig polynomial of any matroid is log-concave. Although the inverse Kazhdan-Lusztig polynomial may not always have only real roots, we conjecture that the Hadamard product of an…

Coxeter matroids generalize matroids just as flag varieties of Lie groups generalize Grassmannians. Valuations of Coxeter matroids are functions that behave well with respect to subdivisions of a Coxeter matroid into smaller ones. We…

In this paper, we introduce a new family of codes relevent for rank and sum-rank metrics. These codes are based on an effective Chinese remainders theorem for linearized polynomials over finite fields. We propose a decoding algorithm for…

This paper is a continuation of my paper "Lattices of flats for symplectic matroids". We explore geometric constructions originating from the lattice of flats of ranked symplectic matroids. We observe that a ranked symplectic matroid always…

In this note, we give a rank function axiomatization for delta-matroids and study the corresponding rank generating function. We relate an evaluation of the rank generating function to the number of independent sets of the delta-matroid,…

We give polynomial-time randomized algorithms for computing the girth and the cogirth of binary matroids that are low-rank perturbations of graphic matroids.

We show that the base polytope $P_M$ of any paving matroid $M$ can be systematically obtained from a hypersimplex by slicing off certain subpolytopes, namely base polytopes of lattice path matroids corresponding to panhandle-shaped Ferrers…

We give a combinatorial interpretation of vector continued fractions obtained by applying the Jacobi-Perron algorithm to a vector of $p\geq 1$ resolvent functions of a banded Hessenberg operator of order $p+1$. The interpretation consists…