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

TL;DR

This paper employs Ehrhart theory from polyhedral geometry to analyze partition congruences, introducing supercranks that combinatorially verify divisibility properties of partitions into three parts for certain primes.

Contribution

It introduces supercranks based on Ehrhart theory to witness partition divisibility congruences, providing explicit bijections and new combinatorial decompositions.

Findings

Supercranks verify divisibility of p(n,3) by primes ≡ -1 mod 6.

Explicit bijections partition sets into m equinumerous classes.

Discussion of behavior for primes ≡ 1 mod 6.

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.