A note on some inequalities for the Tutte polynomial of a matroid

We prove that if a matroid M contains two disjoint bases (or, dually, if its ground set is the union of two bases), then T M ( a , a ) ⩽ max { T M ( 2 a , 0 ) , T M ( 0 , 2 a ) } for a ⩾ 2 . This resembles the conjecture that appears in C. Merino and D.J.A. Welsh, Forests, colourings and acyclic orientations of the square lattice, Annals of Combinatorics 3 (1999) pp. 417–429: If G is a 2-connected graph with no loops, then T G ( 1 , 1 ) ⩽ max { T G ( 2 , 0 ) , T ( 0 , 2 ) } . We conjecture that T M ( 1 , 1 ) ⩽ { T M ( 2 , 0 ) , T M ( 0 , 2 ) } for matroids which contains two disjoint bases or its ground set is the union of two bases. We also prove the latter for some families of graphs and matroids.