Bull. Korean Math. Soc. 2023; 60(4): 1017-1024
Online first article May 17, 2023 Printed July 31, 2023
https://doi.org/10.4134/BKMS.b220457
Copyright © The Korean Mathematical Society.
John Maxwell Campbell
York University
A celebrated result in the study of integer partitions is the identity due to Lehmer whereby the number of partitions of $n$ with an even number of even parts minus the number of partitions of $n$ with an odd number of even parts equals the number of partitions of $n$ into distinct odd parts. Inspired by Lehmer's identity, we prove explicit formulas for evaluating generating functions for sequences that enumerate integer partitions of fixed width with an even/odd number of even parts. We introduce a technique for decomposing the even entries of a partition in such a way so as to evaluate, using a finite sum over $q$-binomial coefficients, the generating function for the sequence of partitions with an even number of even parts of fixed, odd width, and similarly for the other families of fixed-width partitions that we introduce.
Keywords: Partition, parity, generating function, $q$-binomial coefficient
MSC numbers: Primary 11P81, 05A17
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