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.
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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Research in Number Theory