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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 TM(x, y)(More)
The cone Ĝ 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 of Ĝ 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)
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)
Let D be a DAG and let X be any non-empty subset of D’s vertices. X is a convex set of D if D contains no path that originates in X , then visits one or more vertices not in X , and then re-enters X . This work presents basic convexity algorithms for creating, growing, and shrinking convex sets using two different approaches: predecessor and successor sets,(More)
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)