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arXiv · 2609.33770

The two-block odd partition function

Mircea Merca
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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.

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Mircea Merca. 2026-09-27. The two-block odd partition function. https://doi.org/10.1007/s11139-026-01426-1

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