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We show that the number of partitions of n with alternating sum k such thatthe multiplicity of each part is bounded by 2m+1 equals the number ofpartitions of n with k odd parts such that the multiplicity of each even partis bounded by m. The first proof relies on two formulas with two parametersthat are related to the four-parameter formulas of Boulet. We also give acombinatorial proof of this result by using Sylvester's bijection, whichimplies a stronger partition theorem. For m=0, our result reduces toBessenrodt's refinement of Euler's Theorem. If the alternating sum and thenumber of odd parts are not taken into account, we are led to a connection to ageneralization of Euler's theorem, which can be deduced from a theorem ofAndrews on equivalent upper bound sequences of multiplicities. Analogously, weshow that the number of partitions of n with alternating sum k such that themultiplicity of each even part is bounded by 2m+1 equals the number ofpartitions of n with k odd parts such that the multiplicity of each even partis also bounded by 2m+1. We provide a combinatorial proof as well.

Euler's Partition Theorem with Upper Bounds on Multiplicities