On power ideals of transversal matroids and their "parking functions"

TL;DR

This paper reveals that certain power ideals of transversal matroids are monomial and their basis elements correspond to lattice points in a convex polytope called a generalized permutohedron, linking combinatorics and geometry.

Contribution

It demonstrates that specific power ideals associated with transversal matroids are monomial and identifies their basis elements with lattice points in a generalized permutohedron.

Findings

Power ideals of transversal matroids are monomial.

Basis elements correspond to lattice points in a generalized permutohedron.

Connections to Stanley-Reisner theory and matroid h-vectors.

Abstract

To a vector configuration one can associate a polynomial ideal generated by powers of linear forms, known as a power ideal, which exhibits many combinatorial features of the matroid underlying the configuration. In this note we observe that certain power ideals associated to transversal matroids are, somewhat unexpectedly, monomial. Moreover, the (monomial) basis elements of the quotient ring defined by such a power ideal can be naturally identified with the lattice points of a remarkable convex polytope: a polymatroid, also known as generalized permutohedron. We dub the exponent vectors of these monomial basis elements "parking functions" of the corresponding transversal matroid. We highlight the connection between our investigation and Stanley-Reisner theory, and relate our findings to Stanley's conjectured necessary condition on matroid $h$-vectors.