Arithmetic Properties Of Certain Partition Functions

Let b2a (n) denote the number of2a-regular partitions of n. Suppose j is a positive integer and i is odd. If a∈{1,2}, we show that This improves a result of Ono and Penniston. For a>3odd, we show that For a>4even, we prove that where δ(a)=2a/2-1-2.Let t be a positive integer and a,2t(n) the number of2t-cores of n. We show that for t≥3, j≥3and1≤i≤2j, odd, thenThis improves Chen’s result. We also give a conjecture on the distribution of even values of a2t(n) modulo2j.Lastly we explicitly determine k-tuple partitions with even parts distinct modulo8for k=2,4and get infinite families congruences for these partitions. We also get infinite families congruences for partitions with even parts distinct.