Interpretations for the Tutte polynomials of morphisms of matroids

Abstract

We study interpretations of the Tutte polynomials of set-pointed matroids and matroid perspectives. Suppose that M is a matroid whose ground set E is given with a linear ordering < . We show that the Tutte polynomial of M pointed by A ⊆ E equals ∑ x r ( M / ( A ∪ X ) ) y r ∗ ( M − ( A ∪ Y ) ) z r ( ( M / X ) − Y ) where the sum runs through the set D ( M ; A ; < ) of couples ( X , Y ) such that X ⊆ E ∖ A , Y = ( E ∖ A ) ∖ X , and X ∩ C (resp. Y ∩ C ) differs from { min ( C ) } for each circuit C of M ∗ − A (resp. M − A ). Furthermore, we characterize D ( M ; A ; < ) by a contraction–deletion rule. Analogous results are introduced for the Tutte polynomials of matroid perspectives.