### Abstract

An almost partition identity is an identity for partition numbers that is true asymptotically 100% of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.

Original language | English (US) |
---|---|

Pages (from-to) | 5428-5436 |

Number of pages | 9 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 116 |

Issue number | 12 |

DOIs | |

State | Published - Jan 1 2019 |

### All Science Journal Classification (ASJC) codes

- General

### Cite this

*Proceedings of the National Academy of Sciences of the United States of America*,

*116*(12), 5428-5436. https://doi.org/10.1073/pnas.1820945116

}

*Proceedings of the National Academy of Sciences of the United States of America*, vol. 116, no. 12, pp. 5428-5436. https://doi.org/10.1073/pnas.1820945116

**Almost partition identities.** / Andrews, George E.; Ballantine, Cristina.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Almost partition identities

AU - Andrews, George E.

AU - Ballantine, Cristina

PY - 2019/1/1

Y1 - 2019/1/1

N2 - An almost partition identity is an identity for partition numbers that is true asymptotically 100% of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.

AB - An almost partition identity is an identity for partition numbers that is true asymptotically 100% of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.

UR - http://www.scopus.com/inward/record.url?scp=85063278367&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063278367&partnerID=8YFLogxK

U2 - 10.1073/pnas.1820945116

DO - 10.1073/pnas.1820945116

M3 - Article

C2 - 30833382

AN - SCOPUS:85063278367

VL - 116

SP - 5428

EP - 5436

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 12

ER -