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AN ANALOGUE OF EULER’S IDENTITY AND SPLIT PERFECT PARTITIONS

Part of: Additive number theory; partitions Combinatorics

Published online by Cambridge University Press:  17 December 2018

MEGHA GOYAL Open the ORCID record for MEGHA GOYAL [Opens in a new window]
MEGHA GOYAL*
Affiliation:
Department of Mathematical Sciences, IK Gujral Punjab Technical University Jalandhar, Main Campus, Kapurthala-144603, India email meghagoyal2021@gmail.com
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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].

Keywords

q-seriessplit (n + t)-colour partitionsperfect partitionsgenerating functions

MSC classification

Primary: 05A15: Exact enumeration problems, generating functions
Secondary: 05A17: Partitions of integers 11P81: Elementary theory of partitions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 3 , June 2019 , pp. 353 - 361
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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