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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.
Summary
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.
- 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.
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