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AN ANALOGUE OF EULER’S IDENTITY AND SPLIT PERFECT PARTITIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 3 / June 2019
- Published online by Cambridge University Press:
- 17 December 2018, pp. 353-361
- Print publication:
- June 2019
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- Article
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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].
