Divisibility of an analogue of t-core partition function by powers of primes

2024/05/02 by Talukdar, Pranjal
#05A17 #11F11 #11P83 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2405.05274

Abstract

A partition of a positive integer n is said to be t-core if none of its hook lengths are divisible by t. Recently, two analogues, a_t(n) and b_t(n), of the t-core partition function, c_t(n), have been introduced by Gireesh, Ray and Shivashankar \citegrs and Bandyopadhyay and Baruah \citebb, respectively. In this article, we prove the lacunarity of b_t(n) modulo arbitrary powers of 2 and 3 for t=3^αm where gcd(m,6)=1. For a fixed positive integer k and prime numbers p_i≥ 5, we also study the arithmetic density of b_t(n) modulo p_i^k where t=p₁^a₁⋯ p_m^a_m. We further prove an infinite family of congruences for b₃(n) modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators.

Related

• Arithmetic properties of an analogue of t-core partitions
• Arithmetic density and congruences of t-core partitions
• Infinite families of congruences for 2 and 13-core partitions
• A theorem on the cores of partitions
• Distribution of the partition function modulo m