Euler's Partition Theorem with Upper Bounds on Multiplicities

Abstract

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's bijection, which implies a stronger partition theorem. For $m=0$, our result reduces to Bessenrodt's refinement of Euler'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'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.