RISC RISC Research Institute for Symbolic Computation  
  • @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}