AN ANALOGUE OF EULER’S IDENTITY AND SPLIT PERFECT PARTITIONS

Abstract

We give the generating function of split$(n+t)$-colour partitions and obtain an analogue of Euler’s identity for split$n$-colour partitions. We derive a combinatorial relation between the number of restricted split$n$-colour partitions and the function$\unicode[STIX]{x1D70E}_{k}(\unicode[STIX]{x1D707})=\sum _{d|\unicode[STIX]{x1D707}}d^{k}$. We introduce a new class of split perfect partitions with$d(a)$copies of each part$a$and extend the work of Agarwal and Subbarao [‘Some properties of perfect partitions’,Indian J. Pure Appl. Math 22(9) (1991), 737–743].