The covering number of the elements of a matroid and generalized matrix functions

Let S be a nonempty finite set with cardinality m. Let M be a matroid on S with no loops. The covering number of an element x in S is the smallest positive integer k such that x is a coloop of the union of k copies of M. We investigate connections between the structure of M and the values of the covering numbers of elements of S. Applications to the study of the rank partition and generalized matrix functions are presented.