@techreport{RISC5163,author = {Felix Breuer and Dennis Eichhorn and Brandt Kronholm},
title = {{Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts}},
language = {english},
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.
},
year = {2015},
month = {August},
howpublished = {arXiv },
keywords = {Integer partitions, Polyhedral Geometry, Combinatorics, Freeman Dyson, Ramanujan, Ehrhart, Crank, Generating Function, },
length = {28},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}