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Polyhedral Geometry, Supercranks, and Combinatorial Witnesses of Congruences for Partitions Into Three Parts by Breuer, Felix; Eichhorn, Dennis; Kronholm, Brandt is a scholarly article available to read on EtoBox.

What is Polyhedral Geometry, Supercranks, and Combinatorial Witnesses of Congruences for Partitions Into Three Parts about?

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.

Author

Breuer, Felix; Eichhorn, Dennis; Kronholm, Brandt

Published

2015

Language

EN