Partitions with ``d(a) copies of a''
In an earlier paper, \textit{A. K. Agarwal} and \textit{G. E. Andrews} [J. Comb. Theory, Ser. A 45, 40-49 (1987; Zbl 0618.05003)] discussed Rogers- Ramanujan identities for partitions with ``a copies of a''. The authors in the present paper study a class of partitions using d(a) copies of each part a, where d(a) represents the number of positive divisors of a. Several results including combinatorial identities, a generating function, a recursive formula, an asymptotic formula, and a graphical representation for these newly introduced partitions are given in the paper. Also, an identity for partitions with ``d(a) copies of a'' analogous to the identity considered, for instance, in [Can. J. Math. 34, 194-195 (1982; Zbl 0482.10013)] and [J. Comb. Theory, Ser. A 36, 249-252 (1984; Zbl 0532.05003)] are given. The paper concludes by posing several open problems.
scientific article; zbMATH DE number 4064991
scientific article