A Tutte polynomial inequality for lattice path matroids

Let M be a matroid without loops or coloops and let TM be its Tutte polynomial. In 1999 Merino and Welsh conjectured that max(TM (2, 0), TM (0, 2)) ≥ TM (1, 1) 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 show the validity of the multiplicative conjecture when M is a lattice path matroid. In order to do this, we introduce and study a particular class of lattice path matroids, called snakes. We present a characterization showing that snakes are the only graphic lattice path matroids and provide explicit formulas for the number of trees, acyclic orientations and totally cyclic orientations in this case. Snakes are used as building bricks to establish a stronger inequality implying the above multiplicative conjecture as well as to characterize the cases in which equality holds.