On a Partition Identity of Lehmer

TL;DR

This paper proves new identities relating partition counts with specific parity and part restrictions, extending Lehmer's classical results and employing both analytic and combinatorial techniques.

Contribution

It establishes Beck-type companion identities to Lehmer's partition identity and generalizes Lehmer's results using analytic and combinatorial methods.

Findings

Proved identities linking even/odd parts in partitions to partitions into distinct, odd parts.

Generalized Lehmer's identity with new Beck-type companions.

Used both analytic and combinatorial proofs for the identities.

Abstract

Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called "Beck-type" companions to other identities. In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.