On 2-color partitions where one of the colors is multiples of $7^k$

TL;DR

The paper investigates $p_{1,7^k}(n)$, the count of 2-color partitions where one color only appears in parts multiple of $7^k$, whose generating function is $\sum_{n\ge0} p_{1,7^k}(n) q^n = \frac{1}{f_1 f_{7^k}}$. Using a Huffing operator modulo $7$ and a framework of $7$-adic valuations, it derives progression-specific generating-function identities and recurrences for coefficient vectors, enabling the lifting of congruences to arbitrary powers of $7$. The main results are explicit infinite families of Ramanujan-type congruences modulo $7^k$ for both odd and even powers of $7$ in carefully chosen arithmetic progressions, extending prior work on similar restricted partition functions. The methods combine generating-function techniques, matrix recurrences, and $7$-adic analysis to generalize known congruences and provide a systematic approach for deriving further 7-adic congruences in restricted partition settings.

Abstract

In this work, we investigate the arithmetic properties of $p_{1,7^k}(n)$, which counts 2-color partitions of $n$ where one of the colors appears only in parts that are multiples of $7^k$. By constructing generating functions for $p_{1,7^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type infinite family of congruences modulo powers of $7$.

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