Abstract
The smallest part is a rational function. This result is similar to the closely related case of partitions with fixed differences between largest and smallest parts which has recently been studied through analytic methods by Andrews, Beck, and Robbins. Our approach is geometric: We model partitions with bounded differences as lattice points in an infinite union of polyhedral cones. Surprisingly, this infinite union tiles a single simplicial cone. This construction then leads to a bijection that can be interpreted on a purely combinatorial level.
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Acknowledgments
The authors wish to thank George Andrews and Peter Paule for suggesting the topic of fixed differences and asking if polyhedral methods can show that the generating function for partitions with fixed differences is a rational function. Both authors were supported by Austrian Science Fund (FWF) special research group Algorithmic and Enumerative Combinatorics SFB F50, project number F5006-N15.
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Breuer, F., Kronholm, B. A polyhedral model of partitions with bounded differences and a bijective proof of a theorem of Andrews, Beck, and Robbins. Res. number theory 2, 2 (2016). https://doi.org/10.1007/s40993-015-0033-3
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DOI: https://doi.org/10.1007/s40993-015-0033-3