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

TL;DR

This work analyzes how much the configuration of a matroid (the lattice of cyclic flats with their sizes and ranks) constrains isomorphism. It introduces a bumping construction that produces non‑isomorphic matroids sharing the same configuration, and applies it to lattice path matroids to link configuration‑uniqueness with being fundamental transversal. The paper also provides a complete characterization for lattice path matroids regarding modular initial–final flats, yields enumeration results for non‑mixed diagrams and related diagrams, and shows that almost all lattice path matroids are not configuration‑unique. Finally, it proves that every non‑chain lattice occurs as the lattice of cyclic flats of a non‑configuration‑unique transversal matroid, underscoring the abundance of configurational non‑uniqueness in matroid theory.

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 $L$.

Figures (14)

• Figure 1: Two non-isomorphic matroids $M$ and $M'$ that have the same configuration. Each pair shown in the lattice gives the size and rank of the corresponding cyclic flat.
• Figure 2: The region bounded by two paths $P$ and $Q$ that give rise to a lattice path matroid, along with its lattice of cyclic flats, with the rank of each cyclic flat shown below the flat. Each north step is labeled with the position it would have in any lattice path from $(0,0)$ to $(4,5)$ that uses that step.
• Figure 3: The diagrams representing a lattice path matroid $M$ and its restrictions to $[2,8]$ and $[2,7]$. The restriction to $[3,5]$ is the free matroid $U_{3,3}$ on $[3,5]$, and the restriction to $[2,5]$ is also free.
• Figure 4: The matroid $M$ is transversal, but the matroid $M'$ that results when $x$ bumps $y$ is not transversal. Both matroids have rank $3$.
• Figure 5: The lattice of cyclic flats obtained when $3$ bumps $4$ in the matroid shown in Figure \ref{['fig:runningLPMex']}. In this case, only one set is altered; it is highlighted with boldface.

Theorems & Definitions (46)

• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Corollary 2.7
• Theorem 2.8
• Lemma 2.9
• Lemma 2.10