TY - UNPB
T1 - A Polyhedral Model of Partitions with Bounded Differences and a Bijective Proof of a Theorem of Andrews, Beck, and Robbins
AU - Breuer, Felix
AU - Kronholm, James Brandt
PY - 2015
Y1 - 2015
N2 - The main result of this paper is a bijective proof showing that the generating function for partitions with bounded differences between largest and 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.
AB - The main result of this paper is a bijective proof showing that the generating function for partitions with bounded differences between largest and 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.
UR - http://arxiv.org/abs/1505.00250
U2 - 10.48550/arXiv.1505.00250
DO - 10.48550/arXiv.1505.00250
M3 - Preprint
T3 - RISC Report Series
BT - A Polyhedral Model of Partitions with Bounded Differences and a Bijective Proof of a Theorem of Andrews, Beck, and Robbins
PB - RISC
CY - RISC Hagenberg
ER -