On Euler's Theorem

2025/11/06 by Andrews, George E., Kumar, Rahul, Yee, Ae Ja
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.03979

Abstract

Euler's theorem asserts that A(n)=B(n) where A(n) is the number of partitions of n into distinct parts and B(n) is the number of partitions of n into odd parts. In this paper, it is proved that for n>0, A(n)=B(n)=C(n+1)=(1)/(2)D(n+1), where C(n) is the number of partitions of n with largest part even and parts not exceeding half of the largest part are distinct, and D(n) is the number of partitions of n into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.

Related

• Combinatorial proofs of two Euler type identities due to Andrews
• On the partitions into distinct parts and odd parts
• Euler's Partition Theorem with Upper Bounds on Multiplicities
• Generalizations of Euler's Theorem to k-regular partitions
• Combinatorial meaning of the number of the even parts in a partition of n into distinct parts