## Proof of the Alder-Andrews conjecture

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- by Claudia Alfes, Marie Jameson and Robert J. Lemke Oliver PDF
- Proc. Amer. Math. Soc.
**139**(2011), 63-78 Request permission

## Abstract:

Motivated by classical identities of Euler, Schur, and Rogers and Ramanujan, Alder investigated $q_d(n)$ and $Q_d(n),$ the number of partitions of $n$ into $d$-distinct parts and into parts which are $\pm 1 (\operatorname {mod}d+3)$, respectively. He conjectured that $q_d(n) \geq Q_d(n).$ Andrews and Yee proved the conjecture for $d = 2^s-1$ and also for $d \geq 32.$ We complete the proof of Andrews’s refinement of Alder’s conjecture by determining effective asymptotic estimates for these partition functions (correcting and refining earlier work of Meinardus), thereby reducing the conjecture to a finite computation.## References

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## Additional Information

**Claudia Alfes**- Affiliation: Lehrstuhl A für Mathematik, RWTH Aachen, Templergraben 64, D-52062 Aachen, Germany
- Address at time of publication: Fachbereich Mathematik, Technische Universität Darmstadt, Schlossegartenstrasse 7, D-64289 Darmstadt, Germany
- Email: claudia.alfes@matha.rwth-aachen.de
**Marie Jameson**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Address at time of publication: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
- MR Author ID: 913196
- ORCID: 0000-0003-0879-2826
- Email: marie.jameson@gmail.com
**Robert J. Lemke Oliver**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- Address at time of publication: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
- MR Author ID: 894148
- Email: lemkeoliver@gmail.com
- Received by editor(s): March 10, 2010
- Published electronically: July 19, 2010
- Communicated by: Kathrin Bringmann
- © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**139**(2011), 63-78 - MSC (2010): Primary 11P82, 11P84
- DOI: https://doi.org/10.1090/S0002-9939-2010-10500-2
- MathSciNet review: 2729071