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Almost partition identities.

George E Andrews1, Cristina Ballantine2

  • 1Department of Mathematics, Pennsylvania State University, University Park, PA 16802; gea1@psu.edu cballant@holycross.edu.

Proceedings of the National Academy of Sciences of the United States of America
|March 6, 2019
PubMed
Summary
This summary is machine-generated.

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We discovered an "almost partition identity" where self-conjugate partitions of n often match specific restricted partitions. This identity, however, fails infinitely often with growing errors.

Area of Science:

  • Number Theory
  • Combinatorics

Background:

  • Partition identities are fundamental in number theory.
  • Almost partition identities exhibit asymptotic behavior, holding true most of the time but failing infinitely often.

Purpose of the Study:

  • To establish a novel almost partition identity relating self-conjugate partitions to restricted partitions.
  • To explore further identities involving the number of parts in restricted partitions.

Main Methods:

  • Analytic methods to study asymptotic behavior.
  • Combinatorial proofs to provide structural understanding.

Main Results:

  • Proved an almost partition identity: the number of parts in self-conjugate partitions of n is asymptotically equal to partitions of n with no repeated odd parts and exactly one even part.
Keywords:
asymptoticsidentitiespartitions

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  • Demonstrated that this identity fails infinitely often, with unbounded error.
  • Established another identity concerning self-conjugate partitions, distinct odd parts, and specific restricted partitions.

    • Conclusions:

    • The study introduces new insights into partition identities, highlighting their asymptotic nature and occasional failures.
    • The findings offer a deeper understanding of the structure of restricted partitions through novel identities.