Arithmetic properties of 5-regular partitions into distinct parts

2024/11/05 by Nayandeep Deka Baruah, Baruah, Nayandeep Deka, Abhishek Sarma +1
Mathematics · #05A17 #11B37 #11P81 #11P83 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2411.02978

openalex publication_date 2024/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A partition is said to be ℓ-regular if none of its parts is a multiple of ℓ. Let b^′₅(n) denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of n. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of b^′₅(n). We provide full characterization of the parity of b^′₅(2n+1), present several congruences modulo 4, and prove that the generating function of the sequence (b^′₅(5n+1)) is lacunary modulo any arbitrary positive powers of 5.

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