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Let M be a finite matroid with rank function r. We will write A ⊆ M when we mean that A is a subset of the ground set of M , and write M | A and M/A for the matroids obtained by restricting M to A, and contracting M on A respectively. Let M * denote the dual matroid to M. (See [1] for definitions). The main theorem is Theorem 1. The Tutte polynomial T M (x,(More)
A real polynomial is called log-concave if its coefficients form a log-concave sequence. We give a new elementary proof of the fact that a product of log-concave polynomials with nonnegative coefficients and no internal zero coefficients is again log-concave. In addition, we show that if the coefficients of the polynomial m∈M (x + m) form a monotone(More)
The conê G of a finite graph G is obtained by adding a new vertex p, called the cone point, and joining each vertex of G to p by a simple edge. We show that the rank of the reduced homology of the independent set complex of the cycle matroid ofˆG is the cardinality of the set of the edge-rooted forests in the base graph G. We also show that there is a basis(More)
For a d-dimensional cell complex Γ with˜H i (Γ) = 0 for −1 i < d, an i-dimensional tree is a non-empty collection B of i-dimensional cells in Γ such that˜H i (B ∪ Γ (i−1)) = 0 and w(B) := | ˜ H i−1 (B ∪ Γ (i−1))| is finite, where Γ (i) is the i-skeleton of Γ. The i-th tree-number is defined k i := B w(B) 2 , where the sum is over all i-dimensional trees. In(More)
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