One-to-one correspondence between interpretations of the Tutte polynomials

Abstract

We study relation between two interpretations of the Tutte polynomial of a matroid perspective M1→M2 on a set E given with a linear ordering <. A well known interpretation uses internal and external activities on a family B(M1,M2) of the sets independent in M1 and spanning in M2. Recently we introduced another interpretation based on a family D(M1,M2;<) of “cyclic bases” of M1→M2 with respect to <. We introduce a one-to-one correspondence between B(M1,M2) and D(M1,M2;<) that also generates a relation between the interpretations of the Tutte polynomial of a matroid perspective and corresponds with duality.