The two-block odd partition function

Abstract

We investigate arithmetic and combinatorial properties of the function $a(n)$, which is the signed number of partitions of $n$ into exactly two distinct part sizes, each occurring an odd number of times. We prove that $a(n)\ge 0$ for all $n$, establishing a positivity phenomenon for a signed partition function arising from a double Lambert series. Furthermore, we show that $a(n)$ satisfies nontrivial divisibility properties modulo $3$. The results reveal unexpected arithmetic regularity in a family of partition functions defined by odd multiplicity constraints and restricted support.

30 pages