Some congruences for $ \ell $-regular partitions with certain restrictions

Abstract

<abstract><p>Let $ {\rm{pod}}_\ell(n) $ and $ {\rm{ped}}_\ell(n) $ denote the number of $ \ell $-regular partitions of a positive integer $ n $ into distinct odd parts and the number of $ \ell $-regular partitions of a positive integer $ n $ into distinct even parts, respectively. Our first goal in this note was to prove two congruence relations for $ {\rm{pod}}_\ell(n) $. Furthermore, we found a formula for the action of the Hecke operator on a class of eta-quotients. As two applications of this result, we obtained two infinite families of congruence relations for $ {\rm pod}_5(n) $. We also proved a congruence relation for $ {\rm{ped}}_\ell(n) $. In particular, we established a congruence relation modulo 2 connecting $ {\rm{pod}}_\ell(n) $ and $ {\rm{ped}}_\ell(n) $.</p></abstract>