The matroidal Merino–Welsh conjectures

Let $M$ be a matroid without loops or coloops, and let $T(M;x,y)$ be its Tutte polynomial. Matroidal Merino–Welsh conjectures. The following inequalities hold:

$$\max\left(T(M;2,0),T(M;0,2)\right)\geq T(M;1,1),$$ $$T(M;2,0)+T(M;0,2)\geq 2\cdot T(M;1,1),$$

and

$$T(M;2,0)\cdot T(M;0,2)\geq T(M;1,1)^2.$$

These conjectures generalize the graphic inequalities through the Tutte-polynomial interpretations of spanning trees and orientations. The first inequality is known for paving matroids, Catalan matroids, and whirls, while the paper proves the multiplicative conjecture for lattice path matroids; the general matroidal conjectures remain open.