Euler's Partition Theorem with Upper Bounds on Multiplicities

TL;DR

This paper establishes new partition identities involving bounds on multiplicities and alternating sums, connecting classical theorems with generalized bounds and providing multiple proofs including combinatorial and formula-based approaches.

Contribution

It introduces novel partition theorems with upper bounds on multiplicities, extending Euler's and Bessenrodt's theorems, with proofs via formulas and bijections.

Findings

Partitions with bounded multiplicities relate to partitions with odd parts under similar bounds.

The results generalize Euler's theorem and Bessenrodt's refinement.

Multiple proofs, including combinatorial and formula-based, validate the theorems.

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 Theorem. If the alternating sum and the number of odd parts are not taken into account, we are led to a connection to a generalization of Euler's 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.