What's A Permutahedron?

Here we see ranked choice voter data on 4 elements, in this case, the candidates for the Minneapolis city council in 2017. Each vote is recorded as a permutation of those four elements, where the candidate numbers are listed in order of preference.

The permutahedron is a graph with vertices that represent each possible permutation, and edges that represent adjacent transpositions -- if the placement of two candidates are swapped. For example, 1342 and 1432 just have the 3 and 4 in the middle swapped, and have an edge between them.

Once we have built our permutahedron, we can visualize the data on it by coloring the vertices by how many votes they have. Just looking at this picture gives some sense of the trends in the data -- we can see a relatively high proportion of votes on the lower square of this permutahedron, where candidates 3 and 4 are in the first 2 positions. However, these trends are hard to quantify, and only give us a general idea of the data. Most of this project was looking for ways to break down and analyze these trends.