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<p>In this note, we provide three new, very short proofs of two interesting congruences for Merca’s partition function <i>a</i>(<i>n</i>), which enumerates integer partitions where the odd parts have multiplicity at most 2. These modulo 2 congruences were first shown elementarily by Sellers. We then frame <i>a</i>(<i>n</i>) into the much broader context of eta-quotients, and suggest how to comprehensively describe its parity behavior. In particular, extensive computations suggest that <i>a</i>(<i>n</i>) is odd precisely 25% of the time.</p>

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Three more proofs of two congruences for Merca’s partition function

  • 期刊论文

  • 发表日期：

    2025年12月17日

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  • Fabrizio Zanello

摘要

In this note, we provide three new, very short proofs of two interesting congruences for Merca’s partition function a(n), which enumerates integer partitions where the odd parts have multiplicity at most 2. These modulo 2 congruences were first shown elementarily by Sellers. We then frame a(n) into the much broader context of eta-quotients, and suggest how to comprehensively describe its parity behavior. In particular, extensive computations suggest that a(n) is odd precisely 25% of the time.