New Ramanujan-type congruences for 4-core partitions

Ying-Ying Gao ¹   ; Su-Ping Cui ²   ; Litao Guo ³   ; Bernard Lishuang Lin ⁴

¹ Public Education Institute JiLin Province Economic Management Cadre College 130012 ChangChun, P.R. China
² Center for Combinatorics, LPMC-TJKLC Nankai University 300071 Tianjin, P.R. China
³ School of Applied Mathematics Xiamen University of Technology 361024 Xiamen, P.R. China
⁴ School of Science Jimei University 361021 Xiamen, P.R. China

Colloquium Mathematicum, Tome 148 (2017) no. 1, pp. 157-164

Résumé

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 $4$.

Détail

DOI : 10.4064/cm6944-5-2016

Keywords: partition called t core none its hook lengths divisible denote number t cores obtain infinite families congruences modulo example prove ell geq geq biggl ell frac cdot ell biggr equiv mod establish three infinite families congruences modulo