Enumerating partitions arising in homotopy theory

Abstract

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show that these counts are interrelated via partial sums. Moreover, we interpret the partitions we study in the language of graded polynomial rings and apply this to the mod 2 2 Steenrod algebra to compute the free rank of certain homology modules in stable homotopy theory.