Abstract
Dyson’s rank function and the Andrews–Garvan crank function famously give combinatorial witnesses for Ramanujan’s partition function congruences modulo 5, 7, and 11. While these functions can be used to show that the corresponding sets of partitions split into 5, 7, or 11 equally sized sets, one may ask how to make the resulting bijections between partitions organized by rank or crank combinatorially explicit. Stanton recently made conjectures which aim to uncover a deeper combinatorial structure along these lines, where it turns out that minor modifications of the rank and crank are required. Here, we prove two of these conjectures. We also provide abstract criteria for quotients of polynomials by certain cyclotomic polynomials to have non-negative coefficients based on unimodality and symmetry. Furthermore, we extend Stanton’s conjecture to an infinite family of cranks. This suggests further applications to other combinatorial objects. We also discuss numerical evidence for our conjectures, connections with other analytic conjectures such as the distribution of partition ranks.
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Acknowledgements
The first author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 101001179). This work was supported by a grant from the Simons Foundation (853830, LR). The third author is also grateful for support from a 2021-2023 Dean’s Faculty Fellowship from Vanderbilt University and to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support. The authors thank David Chan and Ken Ono for useful discussions related to the topic of the paper. Moreover, we thank the referees for carefully reading our paper and making helpful comments. On behalf of all authors, the corresponding author states that there is no conflict of interest and that there is no associated data for this manuscript.
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Bringmann, K., Gomez, K., Rolen, L. et al. Infinite families of crank functions, Stanton-type conjectures, and unimodality. Res Math Sci 9, 37 (2022). https://doi.org/10.1007/s40687-022-00333-3
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DOI: https://doi.org/10.1007/s40687-022-00333-3