Arithmetic Properties of Partition Quadruples With Odd Parts Distinct

Abstract

Let ${\mathrm{pod}}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of ${\mathrm{pod}}_{-4}(n)$ involving the following infinite family of congruences: for any integers $\alpha \ge 1$ and $n\ge 0$, \[\mathrm{pod}_{-4}\Big({{3}^{α+1}}n+\frac{5\cdot {{3}^{α}}+1}{2}\Big)\equiv 0 \pmod{9}.\] We also establish some internal congruences and some congruences modulo 2, 5 and 8 satisfied by ${\mathrm{pod}}_{-4}(n)$.

arXiv comment10 pages

References in corpus (1)

• Arithmetic Properties of Partition Triples With Odd Parts Distinct