Some congruences for e-regular partitions with certain restrictions

Let $ { m{pod}}_ell(n) $ Face Powder and $ { m{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 $ { m{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 5 Piece Full Panel Bedroom congruence relations for $ { m pod}_5(n) $.We also proved a congruence relation for $ { m{ped}}_ell(n) $.

In particular, we established a congruence relation modulo 2 connecting $ { m{pod}}_ell(n) $ and $ { m{ped}}_ell(n) $.

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