Tutte Polynomials Of Matroids Research Articles

A lattice point counting generalisation of the Tutte polynomial

The Tutte polynomial for matroids is not directly applicable to polymatroids. For instance, deletion- contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This polynomial is constructed using lattice point counts in the Minkowski sum of the base polytope of a polymatroid and scaled copies of the standard simplex. We also show that, in the matroid case, our polynomial has coefficients of alternating sign, with a combinatorial interpretation closely tied to the Dawson partition.

Möbius conjugation and convolution formulae

Let P be a locally finite poset with the interval space Int(P), and R be a ring with identity. We shall introduce the Möbius conjugation μ⁎ sending each function f:P→R to an incidence function μ⁎(f):Int(P)→R such that μ⁎(fg)=μ⁎(f)⁎μ⁎(g). Taking P to be the intersection poset of a hyperplane arrangement A, we shall obtain a convolution identity for the number r(A) of regions and the number b(A) of relatively bounded regions, and a reciprocity theorem of the characteristic polynomial χ(A,t) which gives a combinatorial interpretation of the values |χ(A,−q)| for large primes q. Moreover, all known convolution identities on Tutte polynomials of matroids will be direct consequences after specializing the poset P and functions f,g.

Strongly inequivalent representations and Tutte polynomials of matroids

We develop constructive techniques to show that non-isomorphic 3-connected matroids that are representable over a fixed finite field and that have the same Tutte polynomial abound. In particular, for most prime powers q, we construct infinite families of sets of 3-connected matroids for which the matroids in a given set are non-isomorphic, are representable over GF(q), and have the same Tutte polynomial. Furthermore, the cardinalities of the sets of matroids in a given family grow exponentially as a function of rank, and there are many such families.