Fractional partitions and conjectures of Chern–Fu–Tang and Heim–Neuhauser
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- by Kathrin Bringmann, Ben Kane, Larry Rolen and Zack Tripp HTML | PDF
- Trans. Amer. Math. Soc. Ser. B 8 (2021), 615-634
Abstract:
Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern–Fu–Tang and Heim–Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov–Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern–Fu–Tang conjecture and to show the Heim–Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.References
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Additional Information
- Kathrin Bringmann
- Affiliation: Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
- MR Author ID: 774752
- Email: kbringma@math.uni-koeln.de
- Ben Kane
- Affiliation: Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong
- MR Author ID: 789505
- ORCID: 0000-0003-4074-7662
- Email: bkane@hku.hk
- Larry Rolen
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 923990
- ORCID: 0000-0001-8671-8117
- Email: larry.rolen@vanderbilt.edu
- Zack Tripp
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 1124546
- ORCID: 0000-0001-7100-0161
- Email: zachary.d.tripp@vanderbilt.edu
- Received by editor(s): January 11, 2021
- Received by editor(s) in revised form: April 10, 2021
- Published electronically: July 16, 2021
- Additional Notes: Kathrin Bringmann and Ben Kane are the corresponding authors
The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation. The research of the second author was supported by grants from the Research Grants Council of the Hong Kong SAR, China (project numbers HKU 17301317 and 17303618). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101001179) - © Copyright 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Trans. Amer. Math. Soc. Ser. B 8 (2021), 615-634
- MSC (2020): Primary 11P82
- DOI: https://doi.org/10.1090/btran/77
- MathSciNet review: 4287510