David Perkinson

Permutation polytopes and indecomposable elements in permutation groups

Each group G of n × n permutation matrices has a corresponding permutation polytope, P(G) := conv(G) ⊂ Rn × n. We relate the structure of P(G) to the transitivity of G. In particular, we show that if G has t nontrivial orbits, then min{2t, ⌊n/2⌋} is a sharp upper bound on the diameter of the graph of P(G). We also show that P(G) achieves its maximal dimension of (n - 1)2 precisely when G is 2-transitive. We then extend the results of Pak [I. Pak, Four questions on Birkhoff polytope, Ann. Comb. 4 (1) (2000) 83-90] on mixing times for a random walk on P(G). Our work depends on a new result for permutation groups involving writing permutations as products of indecomposable permutations.

G-parking functions and tree inversions

A depth-first search version of Dhar’s burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the spanning tree. Specializing to the complete graph solves a problem posed by R. Stanley.

Sandpiles, Spanning Trees, and Plane Duality

Let

PRIMER FOR THE ALGEBRAIC GEOMETRY OF SANDPILES

G

Curves in Grassmannians

be a connected, loopless multigraph. The sandpile group of

Inflections of toric varieties

G

Some facets of the polytope of even permutation matrices

is a finite abelian group associated to

Eight Lectures on Monomial Ideals

G

Principal parts of line bundles on toric varieties

whose order is equal to the number of spanning trees in

Orientations, semiorders, arrangements, and parking functions

G