Expected value of the minimal basis of random matroid and distributions of q-analogs of order statistics

Abstract

Abstract We formulate and prove a formula to compute the expected value of the minimal random basis of an arbitrary finite matroid whose elements are assigned weights which are independent and uniformly distributed on the interval [0, 1]. This metod yields an exact formula in terms of the Tutte polynomial. We give a simple formula to find the minimal random basis of the projective geometry P G ( r − 1 , q ) . In the second part we give a distribution and asymptotic distributions of q-analogs of the k-th order statistics and the intermediate order statistics with r − k → ∞ when n is a number of elements of the projective geometry P G ( r − 1 , q ) . The proofs will appear in a forthcoming publications.