The two-block odd partition function

Abstract

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 $$a(n)\ge 0$$ a ( n ) ≥ 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.