@**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},

url = {http://arxiv.org/abs/1508.00397},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}