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

2025/09/11 by Fabrizio Zanello, Zanello, Fabrizio
Mathematics · #11P84 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11P83 #Secondary: 05A17

paper · pdf · doi:10.48550/arxiv.2509.09833

openalex publication_date 2025/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice

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