  In recent work with Harman, we introduced a new notion of measure for
oligomorphic groups, and showed how they can be used to produce interesting
tensor categories. Determining the measures for an oligomorphic group is (in
our view) an important and difficult combinatorial problem, which has only been
solved in a handful of cases. The purpose of this paper is to solve this
problem for a certain infinite family of oligomorphic groups, namely, the
automorphism group of the $n$-colored circle (for each $n \ge 1$).
