On $\ell-$regular and $2-$color partition triples modulo powers of $3$

TL;DR

The paper investigates congruences modulo powers of $3$ for two families of partition-Counting functions: $T_ (n)$, the $ - ext{regular}$ partition triples, and $p_{ ,3}(n)$, the $2$-color partition triples with a color restricted to multiples of $ $. It develops a generating-function framework using the $H$ operator and related modular-forms machinery to propagate 3-adic divisibility through coefficient recurrences, yielding infinite families of congruences for $T_{3^{2eta+1}}$, $T_{3^{2eta+2}}$, and their variants in arithmetic progressions. The results cover cases where $ \\equiv 0 mod 3^k$ and $ \\equiv \\pm 3^k mod 3^{k+1}$, including refinements involving primes and representations as sums of squares (e.g., $x^2+3y^2$), and they confirm conjectures BC1 and BC2. The methods extend prior work of Gireesh–Naika and Tang, unifying several 3-adic congruence phenomena for partition triples via generating-functions, $H$-operations, and recursion on coefficient structures. These findings have implications for the modular structure of colored partition statistics and their arithmetic properties in modular arithmetic contexts.

Abstract

Let $T_\ell(n)$ denote the number of $\ell-$regular partition triples of $n$ and let $p_{\ell, 3}(n)$ enumerates the number of 2--color partition triples of $n$ where one of the colors appear only in parts that are multiples of $\ell$. In this paper, we prove several infinite families of congruences modulo powers of 3 for $T_\ell(n)$ and $p_{\ell, 3}(n)$, where $\ell \geq 1$ and $\equiv 0\pmod{3^k}$, and $\equiv \pm 3^k \pmod{3^{k+1}}$.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• Lemma 2.2: H and T
• proof