Lattice path matroids: enumerative aspects and Tutte polynomials

Abstract

Fix two lattice paths P and Q from (0,0) to (m,r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0,0) to (m,r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call a lattice path matroid. We consider a variety of enumerative aspects of these matroids and we study three important matroid invariants, namely the Tutte polynomial and, for special types of lattice path matroids, the characteristic polynomial and the beta invariant. In particular, we show that the Tutte polynomial is the generating function for two basic lattice path statistics and we show that certain sequences of lattice path matroids give rise to sequences of Tutte polynomials for which there are relatively simple generating functions. We show that Tutte polynomials of lattice path matroids can be computed in polynomial time. Also, we obtain a new result about lattice paths from an analysis of the beta invariant of certain lattice path matroids.

Figures (11)

• Figure 1: The bases $\{4,5,6\}$, $\{3,5,6\}$, $\{3,4,6\}$, $\{2,5,6\}$, and $\{2,4,6\}$ of a lattice path matroid represented as the North steps of lattice paths.
• Figure 2: Presentations of a lattice path matroid and its dual.
• Figure 3: Presentations of two lattice path matroids and their direct sum.
• Figure 4: Presentations of the rank nine matroid $M^{2,3}_3$, the $3$-Catalan matroid $M^3_4$ of rank four, and the rank six Catalan matroid $M_6$.
• Figure 5: A parallelogram polyomino and its associated Dyck path.

Theorems & Definitions (58)

• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Definition 3.1
• Definition 3.2
• Theorem 3.3
• proof