Arithmetic Density and Congruences of t-Core Partitions

Article Type

Research Article

Publication Title

Results in Mathematics

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 (Bull Aust Math Soc 1:51–55, 2019) obtained a parity result for 3-core partition function a3(n) . Recently, Meher and Jindal (Arithmetic density and new congruences for 3-core 590 Partitions, 2023) 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, a3αm(n) is almost always divisible by arbitrary power of 2 and 3. Further, we prove that at(n) is almost always divisible by arbitrary power of pij, where j is a fixed positive integer and t=p1a1p2a2…pmam with primes pi≥ 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 ap(n) , where p is prime number. Our results also generalizes the results in Radu and Sellers (Acta Arith 146:43–52, 2011).

DOI

10.1007/s00025-023-02032-z

Publication Date

2-1-2024

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