On a partition identity of Lehmer

• Published in: Discrete mathematics 2022-10, Vol.345 (10), p.112979, Article 112979
• Main Authors: Ballantine, Cristina, Burson, Hannah, Folsom, Amanda, Hsu, Chi-Yun, Negrini, Isabella, Wen, Boya
• Format: Article
• Language: English
• Subjects: Beck-type identities; Lehmer's identity; Partitions; q-series
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2022.112979

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.