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arXiv · 1508.00397

Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts

Felix Breuer·Dennis Eichhorn·Brandt Kronholm
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Abstract

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.

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Felix Breuer, Dennis Eichhorn, Brandt Kronholm. 2015-08-03. Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts. https://arxiv.org/abs/1508.00397

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