Arithmetic Properties of Partitions with 1-colored Even Parts and r-colored Odd Parts
M. P. Thejitha, S. N. Fathima
TL;DR
The paper studies the $r$-colored partition function $a_r(n)$, counting partitions of $n$ with even parts in one color and odd parts in $r$ colors, with generating function $\sum_{n\ge0} a_r(n) q^n = \dfrac{f_2^{r-1}}{f_1^r}$. By combining Newman’s modular-form result with eta-quotients and Sturm’s bound, the authors derive extensive infinite families of congruences modulo $3$ and $5$, extending prior results for $a_r(n)$. They prove modulo-$5$ congruences for $a_3(n)$ and modulo-$3$ congruences for $a_t(n)$ with $t$ in a large set, and also obtain combined modulo-$3$ and modulo-$5$ congruences for $a_5(n)$ using Sturm-type arguments and explicit modular forms, highlighting the power of modular forms in partition arithmetic.
Abstract
Recently, Hirschhorn and Sellers defined the partition function $a_r(n)$, which counts the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may appear in one of $r$-colors for fixed $r\ge1$. The aim of this paper is to prove several new infinite families of congruences modulo 3 and 5 by employing a result of Newman and theory of modular forms.