New Ramanujan-type congruences for 4-core partitions
Tom 148 / 2017
Colloquium Mathematicum 148 (2017), 157-164
MSC: Primary 05A17; Secondary 11P83.
DOI: 10.4064/cm6944-5-2016
Opublikowany online: 3 March 2017
Streszczenie
A partition is called a $t$-core if none of its hook lengths is divisible by $t$. Let $a_t(n)$ denote the number of $t$-cores of $n$. We obtain two infinite families of congruences modulo $5$ for $a_4(n)$. For example, we prove that for $\ell\geq 1$ and $n\geq 0$, $$ a_4\biggl(5^{2\ell+1}n+\frac{21\cdot5^{2\ell}-5}{8}\biggr)\equiv 0\ ({\rm mod} 5). $$ We also establish three infinite families of congruences modulo $4$.
