Fibonacci numbers, alternating parity sequences and faces of the tridiagonal Birkhoff polytope

Abstract

We determine the number of alternating parity sequences that are subsequences of an increasing m-tuple of integers. For this and other related counting problems we find formulas that are combinations of Fibonacci numbers. These results are applied to determine, among other things, the number of vertices of any face of the polytope of tridiagonal doubly stochastic matrices.