Internal Congruences Modulo Powers of 5 for Partition k-Tuples with Odd Parts Distinct

Abstract

Let pod − k ( n ) denote the number of partition k-tuples of n where the odd parts in each partition are distinct. By utilizing some q-series manipulations and iterative computations, we derive dozens of internal congruences modulo powers of 5 for pod − k ( n ) with 1 ≤ k ≤ 4 . For example, one result proved in the present paper is that for any n ≥ 0 and 2 ≤ m ≤ 12 , pod − 4 ( 5 N m n + 5 N m + 1 2 ) ≡ pod − 4 ( 5 m n + 5 m + 1 2 ) ( mod 5 m ) , where N m = 2 × 5 m − 2 + m . Further, we conjecture that these internal congruences exist in the corresponding internal congruence families modulo any powers of 5.