Enumerative Problems of Doubly Stochastic Matrices and the Relation to Spectra

Abstract

This work concerns the spectra of doubly stochastic matrices whose entries are rational numbers with a bounded denominator. When the bound is fixed, we consider the enumeration of these matrices and also the enumeration of the orbits under the action of the symmetric group. In the case where the bound is two, we investigate the symmetric case. Such matrices are in fact doubly stochastic, and have a nice characterization when we are in the special case where the diagonal is zero. As a central tool to this investigation, we utilize Birkhoff's theorem that asserts that the doubly stochastic matrices are exactly the polytope defined by the convex hull of permutation matrices. In particular, we consider the spectra along segments in the Birkhoff polytope.