Infinite families of congruences for 2 and 13-core partitions

2023/02/23 by Jindal, Ankita, Meher, Nabin Kumar
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2302.12257

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). Motivated by this result, both the authors \citeMeherJindal2022 recently proved that for a non-negative integer α, a_{3^α m}(n) is almost always divisible by arbitrary power of 2 and 3 and 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. In this article, by using Hecke eigenform theory, we obtain infinite families of congruences and multiplicative identities for a₂(n) and a₁₃(n) modulo 2 which generalizes some results of Das \citeDas2016.

Related

• Arithmetic density and congruences of t-core partitions
• Divisibility of an analogue of t-core partition function by powers of primes
• Arithmetic properties of an analogue of t-core partitions
• Congruences of the partition function
• Arithmetic Identities and Congruences for Partition Triples with 3-cores