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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Partition theorems for Euler pairs


Author: M. V. Subbarao
Journal: Proc. Amer. Math. Soc. 28 (1971), 330-336
MSC: Primary 10.48
DOI: https://doi.org/10.1090/S0002-9939-1971-0274410-1
MathSciNet review: 0274410
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Abstract: This paper generalizes and extends the recent results of George Andrews on Euler pairs. If $ {S_1}$ and $ {S_2}$ are nonempty sets of natural numbers, we define $ ({S_1},{S_2})$ to be an Euler pair of order $ r$ whenever $ {q_r}({S_1};n) = p({S_2};n)$ for all natural numbers $ n$, where $ {q_r}({S_1};n)$ denotes the number of partitions of $ n$ into parts taken from $ {S_1}$, no part repeated more than $ r - 1$ times $ (r > 1)$, and $ p({S_2};n)$ the number of partitions of $ n$ into parts taken from $ {S_2}$. Using a method different from Andrews', we characterize all such pairs, and consider various applications as well as an extension to vector partitions.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0274410-1
Keywords: Partition, generating function, prime number, quadratic and higher power residues, vector partition
Article copyright: © Copyright 1971 American Mathematical Society

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