Congruences for generalized Color Partitions of Hirschhorn and Sellers

TL;DR

The article investigates Ramanujan-type congruences for generalized color partitions $a_k(n)$, where even parts are monochromatic and odd parts carry $k$ colors, with generating function $\sum_{n\ge0} a_k(n) q^n = \frac{f_2^{k-1}}{f_1^k}$. Employing an elementary $q$-series approach based on Ramanujan theta functions, Jacobi's triple product, and dissections, it establishes infinite families of congruences modulo $2$, $2^k$, and $11$ for various $k$, and derives a recurrence linking $a_k(n)$ to ordinary partitions and colored distinct partitions. Key results include $a_{2^r}(2n+1) \equiv 0 \pmod{2^r}$, $a_{2k}(2n+1) \equiv 0 \pmod{2}$, and $a_{11j+k}(11n+r) \equiv 0 \pmod{11}$ for several residues, along with a recurrence formula $\sum a_k(n) q^n = (\frac{f_2}{f_1})^{k-1} \frac{1}{f_1}$. The work broadens the understanding of Ramanujan-type congruences in colored partition settings and provides groundwork for future elementary proofs of higher modulus congruences.

Abstract

Let $a_k(n)$ denote the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may be ``colored" with one of $k$ colors, for fixed $k$. In this note, we find some congruences for $a_k(n)$ in the spirit of Ramanujan's congruences. We prove a number of results for $a_k(n)$ modulo 2, $2^k$ and 11. We also obtain a recurrence relation for $a_k(n)$. Our approach is truly elementary, relying on $q$-dissection techniques.

Theorems & Definitions (28)

• Theorem 1.1: r4,Corollary 3.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: r8, corollary 3.1
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 1: Euler's Pentagonal number Theorem (r3, Entry 22)