  This paper gives a novel and compact proof that a metric graph consisting of
a chain of loops of torsion order $0$ is Brill-Noether general (a theorem of
Cools-Draisma-Payne-Robeva), and a finite or metric graph consisting of a chain
of loops of torsion order $k$ is Hurwitz-Brill-Noether general in the sense of
splitting loci (a theorem of Cook-Powell-Jensen). In fact, we prove a
generalization to (metric) graphs with two marked points, that behaves well
under vertex gluing. The key construction is a way to associate permutations to
divisors on twice-marked graphs, simultaneously encoding the ranks of every
twist of the divisor by the marked points. Vertex gluing corresponds to the
Demazure product, which can be formulated via tropical matrix multiplication.

Combinatorics;Algebraic Geometry