A Tutte polynomial inequality for lattice path matroids

TL;DR

This paper proves a multiplicative inequality involving the Tutte polynomial for a specific class of matroids called lattice path matroids, extending previous results and characterizing equality cases.

Contribution

It establishes the multiplicative conjecture for lattice path matroids and introduces snakes as fundamental structures for this class.

Findings

Proved the multiplicative Tutte polynomial inequality for lattice path matroids.

Characterized cases of equality in the inequality.

Extended known results from uniform and Catalan matroids.

Abstract

Let $M$ be a matroid without loops or coloops and let $T(M;x,y)$ be its Tutte polynomial. In 1999 Merino and Welsh conjectured that $$\max(T(M;2,0), T(M;0,2))\geq T(M;1,1)$$ holds for graphic matroids. Ten years later, Conde and Merino proposed a multiplicative version of the conjecture which implies the original one. In this paper we prove the multiplicative conjecture for the family of lattice path matroids (generalizing earlier results on uniform and Catalan matroids). In order to do this, we introduce and study particular lattice path matroids, called snakes, used as building bricks to indeed establish a strengthening of the multiplicative conjecture as well as a complete characterization of the cases in which equality holds.