A Generalization to Bases Common to r Binary Matroids A the Weighted Matrix-Tree Theorem in the Case when the Weights are Boolean

Abstract

Let M 1 , M 2 , …, M r be r binary matroids of the same rank on the same set S = { s 1 , s 2 , …, s m }. Associate with each element s i of S the weight x i where x i ∈ GF(2), the field of the integers mod 2. Set x = ( x 1 , x 2 , …, x m ). Let ℬ denote the set of bases common to all these matroids and consider the polynomial β(x) = ∑ b ∈ ℬ ∏ s i ∈ b x i . In this paper we obtain a formula for β(x) in the form of the permanent of an r -dimensional array or ‘ r -dimensional tensor’. The weighted matrix-tree theorem and the weighted matrix-arborescence theorem (with Boolean weights) are both special cases of this result in which two matroids are involved ( r = 2). Employing the main formula with r = 2 we show that M 1 and M 2 have an even number of bases in common if and only if there exists a subset P of S such that P can be partitioned into circuits of M 1 and P can be partitioned into cocircuits of M 2 .