NEW CONGRUENCES AND DENSITY RESULTS FOR t-REGULAR PARTITIONS WITH DISTINCT EVEN PARTS

Let t≥2 be a fixed positive integer. Let pedt(n) denote the number of t-regular partitions of n wherein the even parts are distinct and the odd parts are unrestricted. We establish infinite families of congruences for pedt(n) modulo certain positive integers M, for specific values of t. We next study the distribution of pedt(n) for t=3,5,7,9. We prove that the series ∑n=0∞pedt(2n+1)qn is lacunary modulo arbitrary powers of 2 for t=3,5,9. We also prove that the series ∑n=0∞ped7(2n+1)qn is lacunary modulo 2. We use arithmetic properties of modular forms and Hecke eigenforms to prove our results.