On $7^k$-regular partitions modulo powers of $7$

TL;DR

The paper extends Ramanujan-type congruences from 7-regular partitions to all $7^k$-regular partitions by constructing a 7-adic generating-function framework. Using the Huffing operator and a coefficient-matrix $M=(m_{i,j})$, it derives progression-specific generating functions for $b_{7^{2k-1}}$ and $b_{7^{2k}}$ expressed via powers of $(f_7/f_1)$ and recursively defined coefficient vectors. Through careful 7-adic valuation analysis of these coefficient vectors, the authors prove two general families of congruences: $b_{7^{2k-1}}(7^{2k+2\beta-1}n+\frac{18\cdot7^{2k+2\beta-1}-7^{2k-1}+1}{24})\equiv 0\pmod{7^{k+\beta}}$ and $b_{7^{2k}}(7^{2k+\beta-1}(n+1)-\frac{7^{2k}-1}{24})\equiv 0\pmod{7^{k+\beta}}$. These results generalize previous congruences (e.g., for $k=1$) and provide a unified generating-function approach for all powers of 7 in the regular-partition setting, highlighting 7-adic structure in partition arithmetic.

Abstract

In this study, we explore the arithmetic properties of $b_{7^k}(n)$ for any $k\geq1$, which enumerates the partitions of $n$ where no part is divisible by $7^k$. By constructing generating functions for $b_{7^k}(n)$ over specific arithmetic progressions, we establish a collection of Ramanujan-type congruences.

Theorems & Definitions (14)

• Theorem 1.1
• Lemma 2.1: FG, Lemma 3.1 and 3.4
• Lemma 2.2
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Lemma 4.1: FG, Lemma 5.1
• Lemma 4.2: FG, Lemma 5.3