A construction that preserves the configuration of a matroid, with applications to lattice path matroids

TL;DR

This paper introduces a construction that alters a matroid's configuration without changing its invariants, explores its application to lattice path matroids, and discusses conditions under which matroids are uniquely determined by their configuration.

Contribution

It presents a method to produce non-isomorphic matroids with identical configurations and invariants, and characterizes when lattice path matroids are uniquely determined by their configuration.

Findings

The construction applies to non-fundamental transversal lattice path matroids.

Most lattice path matroids are not Tutte unique asymptotically.

Conditions are provided for matroids to be determined by their configuration.

Abstract

The configuration of a matroid $M$ is the abstract lattice of cyclic flats (flats that are unions of circuits) where we record the size and rank of each cyclic flat, but not the set. One can compute the Tutte polynomial of $M$, and stronger invariants (notably, the $\mathcal{G}$-invariant), from the configuration. Given a matroid $M$ in which certain pairs of cyclic flats are non-modular, we show how to produce a matroid that is not isomorphic to $M$ but has the same configuration as $M$. We show that this construction applies to a lattice path matroid if and only if it is not a fundamental transversal matroid, and we enumerate the connected lattice path matroids on $[n]$ that are fundamental; these results imply that, asymptotically, almost no lattice path matroids are Tutte unique. We give a sufficient condition for a matroid to be determined, up to isomorphism, by its configuration. We treat constructions that yield matroids with different configurations where each matroid is determined by its configuration and all have the same $\mathcal{G}$-invariant. We also show that for any lattice $L$ other than a chain, there are non-isomorphic transversal matroids that have the same configuration and where the lattices of cyclic flats are isomorphic to $L$.