TY - UNPB
T1 - Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts
AU - Breuer, Felix
AU - Eichhorn, Dennis
AU - Kronholm, James Brandt
PY - 2015/8
Y1 - 2015/8
N2 - In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn allows us to define statistics called {\it supercranks} that combinatorially witness every instance of divisibility of $p(n,3)$ by any prime $m \equiv -1 \pmod 6$, where $p(n,3)$ is the number of partitions of $n$ into three parts. A rearrangement of lattice points allows us to demonstrate with explicit bijections how to divide these sets of partitions into $m$ equinumerous classes. The behavior for primes $m' \equiv 1 \pmod 6$ is also discussed.
AB - In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn allows us to define statistics called {\it supercranks} that combinatorially witness every instance of divisibility of $p(n,3)$ by any prime $m \equiv -1 \pmod 6$, where $p(n,3)$ is the number of partitions of $n$ into three parts. A rearrangement of lattice points allows us to demonstrate with explicit bijections how to divide these sets of partitions into $m$ equinumerous classes. The behavior for primes $m' \equiv 1 \pmod 6$ is also discussed.
UR - http://arxiv.org/abs/1508.00397
U2 - 10.48550/arXiv.1508.00397
DO - 10.48550/arXiv.1508.00397
M3 - Preprint
T3 - RISC Report Series
BT - Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts
PB - RISC
CY - RISC Hagenberg
ER -