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

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(5Nmn+5Nm+12)≡pod−4(5mn+5m+12) (mod5m), where Nm=2×5m−2+m. Further, we conjecture that these internal congruences exist in the corresponding internal congruence families modulo any powers of 5.