## 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.

Original language | English (US) |
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Article number | 112979 |

Journal | Discrete Mathematics |

Volume | 345 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2022 |

### Bibliographical note

Funding Information:The authors thank the Banff International Research Station (BIRS) and the Women in Numbers 5 (WIN5) Program. The third author is partially supported by National Science Foundation Grant DMS-1901791. The fifth author is partially supported by a FRQNT scholarship by Fonds de recherche du Québec, and an ISM scholarship by Institut des Sciences Mathématiques.

Funding Information:

The authors thank the Banff International Research Station (BIRS) and the Women in Numbers 5 (WIN5) Program. The third author is partially supported by National Science Foundation Grant DMS-1901791 . The fifth author is partially supported by a FRQNT scholarship by Fonds de recherche du Québec , and an ISM scholarship by Institut des Sciences Mathématiques .

Publisher Copyright:

© 2022 Elsevier B.V.

## Keywords

- Beck-type identities
- Lehmer's identity
- Partitions
- q-series