Enumerating partitions arising in homotopy theory

TL;DR

This paper develops recursive formulas for counting special binary partitions, interprets them in algebraic terms, and applies these results to compute homology module ranks in stable homotopy theory.

Contribution

It introduces new recursive formulas for binary partitions with divisibility conditions and connects these counts to algebraic structures in homotopy theory.

Findings

Derived recursive formulas for counting partitions.

Connected partition counts to algebraic structures in homotopy theory.

Computed free ranks of homology modules using these formulas.

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 $2$ Steenrod algebra to compute the free rank of certain homology modules in stable homotopy theory.