Arithmetic properties of partitions with designated summands.

A new class of partitions, partitions with designated summands, was introduced by Andrews, Lewis and Lovejoy. Let PD ( n ) denote the number of partitions of n with designated summands. Andrews, Lewis and Lovejoy established many congruences modulo 3 and powers of 2 for PD ( n ) by using the theory of modular forms. In this paper, we prove several infinite families of congruences modulo 9 and 27 for PD ( n ) by employing the generating functions of PD ( 3 n ) and PD ( 3 n + 1 ) which were discovered by Chen, Ji, Jin and Shen. For example, we prove that for n ≥ 0 and k ≥ 1 , PD ( 2 18 k − 1 ( 12 n + 1 ) ) ≡ 0 ( mod 27 ) . Furthermore, using some results due to Newman, we find some strange congruences modulo 27 for PD ( n ) . For example, we prove that for k ≥ 0 , PD ( 13 9 k ( 75 p + 2 ) ) ≡ 0 ( mod 27 ) and PD ( 2 × 13 9 k + 8 ) ≡ 0 ( mod 27 ) , where p is a prime and p ≡ 1 ( mod 12 ) . [ABSTRACT FROM AUTHOR]