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Partition Theorems for Euler Pairs

M. V. Subbarao
Proceedings of the American Mathematical Society
Vol. 28, No. 2 (May, 1971), pp. 330-336
DOI: 10.2307/2037963
Stable URL: http://www.jstor.org/stable/2037963
Page Count: 7
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Partition Theorems for Euler Pairs
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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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