Some congruences for 3-component multipartitions (Q330344)

A multipartition of \(n\) with \(r\)-components is an \(r\)-tuple \(\lambda=(\lambda^{(1)}, \lambda^{(2)}, \ldots, \lambda^{(r)})\) of partitions whose weights sum to \(n\), it is also referred to as an \(r\)-colored partition. Let \(p_r(n)\) denote the number of \(r\)-component multipartitions of \(n\). The generating function of \(p_r(n)\) is \(\sum\limits_{n=0}^{\infty}p_r(n)q^n=\frac{1}{(q,\,q)_{\infty}^r}\).NEWLINENEWLINEThe paper under review is concerned with congruence properties of \(p_3(n)\) modulo powers of 3. The authors obtain eight congruences for \(p_3(n)\) modulo powers of 3. For example, they prove that for \(n\geq 0\), NEWLINE\[NEWLINEp_3(243n+233)\equiv 0\pmod{3^{10}},NEWLINE\]NEWLINE NEWLINE\[NEWLINEp_3(729n+638)\equiv 0\pmod{ 3^{10}}.NEWLINE\]NEWLINENEWLINENEWLINEThe proofs are based on some theta function identities.