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	<title>Euler&#039;s Partition Theorem with Upper Bounds on Multiplicities | Chen | The Electronic Journal of Combinatorics</title>
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	<meta name="DC.Creator.PersonalName" content="William Y.C. Chen"/>
	<meta name="DC.Creator.PersonalName" content="Ae Ja Yee"/>
	<meta name="DC.Creator.PersonalName" content="Albert J. W. Zhu"/>
	<meta name="DC.Date.created" scheme="ISO8601" content="2012-10-04"/>
	<meta name="DC.Date.dateSubmitted" scheme="ISO8601" content="2012-05-06"/>
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	<meta name="DC.Description" xml:lang="en" content=" We show that the number of partitions of $n$ with alternating sum $k$ such that the multiplicity of each part is bounded by $2m+1$ equals the number of partitions of $n$ with $k$ odd parts such that the multiplicity of each even part is bounded by $m$. The first proof relies on two formulas with two parameters that are related to the four-parameter formulas of Boulet. We also give a combinatorial proof of this result by using Sylvester&#039;s bijection, which implies a stronger partition theorem. For $m=0$, our result reduces to Bessenrodt&#039;s refinement of Euler&#039;s partition theorem. If the alternating sum and the number of odd parts are not taken into account, we are led to a generalization of Euler&#039;s partition theorem, which can be deduced from a theorem of Andrews on equivalent upper bound sequences of multiplicities. Analogously, we show that the number of partitions of $n$ with alternating sum $k$ such that the multiplicity of each even part is bounded by $2m+1$ equals the number of partitions of $n$ with $k$ odd parts such that the multiplicity of each even part is also bounded by $2m+1$. We provide a combinatorial proof as well. "/>
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						<meta name="DC.Subject" xml:lang="en" content="partition, Euler&#039;s partition theorem, Sylvester&#039;s bijection"/>
				<meta name="DC.Title" content="Euler&#039;s Partition Theorem with Upper Bounds on Multiplicities"/>
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