Arithmetic properties of partitions with designated summands
Let \(PD(n)\) denote the number of partitions of \(n\) with designated summands. \textit{W. Y. C. Chen} et al. [J. Number Theory 133, No. 9, 2929--2938 (2013; Zbl 1295.05044)] established the generating functions \[ \sum_{n \geq 0} PD(3n+2)q^n = 3\frac{f_3^3f_6^3}{f_1^5f_2^3} \] and \[ \sum_{n \geq 0} PD(3n)q^n = \frac{f_3^3}{f_1^5} \left(\frac{f_4^6f_6}{f_2^5f_{12}^2} + 3q^2\frac{f_{12}^6}{f_2f_4^2f_{6}^3}\right), \] where \[ f_k := \prod_{n \geq 1} (1-q^{kn}). \] With these identities as a starting point, the author proves a number of congruences for \(PD(n)\) modulo \(9\) or \(27\). Among them are an exact formula for \(PD(36n+9)\) modulo \(9\) which depends on the factorization of \(8n+2\), families of Ramanujan-type congruences such as \[ PD(2^{6k-1}(12n+1)) \equiv 0 \pmod{9}, \] and a congruence involving primes \(p \equiv 1 \pmod{12}\), \[ PD(13^{9k}(75p+2)) \equiv 0 \pmod{27}. \]
scientific article; zbMATH DE number 6497372