Arithmetic density and congruences of t-core partitions

2023/02/23 by Nabin Kumar Meher, Meher, Nabin Kumar, Ankita Jindal +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2302.11830

openalex publication_date 2023/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A partition of n is called a t-core partition if none of its hook number is divisible by t. In 2019, Hirschhorn and Sellers \citeHirs2019 obtained a parity result for 3-core partition function a₃(n). Recently, both authors \citeMeherJindal2022 proved density results for a₃(n), wherein we proved that a₃(n) is almost always divisible by arbitrary power of 2 and 3. In this article, we prove that for a non-negative integer α, a_{3^α m}(n) is almost always divisible by arbitrary power of 2 and 3. Further, we prove that a_t(n) is almost always divisible by arbitrary power of p_i^j, where j is a fixed positive integer and t= p₁^a₁p₂^a₂… p_m^a_m with primes p_i ≥ 5. Furthermore, by employing Radu and Seller's approach, we obtain an algorithm and we give alternate proofs of several congruences modulo 3 and 5 for a_p(n), where p is prime number. Our results also generalizes the results in \citeradu2011a.

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