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)≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(n)\ge 0$$\end{document} 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.