Formulas for Partition $k$-Tuples with $t$-Cores

TL;DR

This paper derives formulas for counting partition $k$-tuples with $t$-core partitions using modular forms, extending previous results for specific cases.

Contribution

It introduces new formulas for $A_{t,k}(n)$ for certain $t$ and $k$ values, expanding the understanding of $t$-core partition tuples.

Findings

Formulas for $A_{t,k}(n)$ for specific $t$ and $k$ values

Extension of Wang's results for $t=3$

Application of modular forms method

Abstract

Let $A_{t,k}(n)$ denote the number of partition $k$-tuples of $n$ where each partition is $t$-core. In this paper, we establish formulas of $A_{t,k}(n)$ for some values of $t$ and $k$ by employing the method of modular forms, which extends Wang's result for $t=3$ and $k=2,3$.