  Elliptic functions considered by Dixon in the nineteenth century and related
to Fermat's cubic, $x^3+y^3=1$, lead to a new set of continued fraction
expansions with sextic numerators and cubic denominators. The functions and the
fractions are pregnant with interesting combinatorics, including a special
P\'olya urn, a continuous-time branching process of the Yule type, as well as
permutations satisfying various constraints that involve either parity of
levels of elements or a repetitive pattern of order three. The combinatorial
models are related to but different from models of elliptic functions earlier
introduced by Viennot, Flajolet, Dumont, and Fran{\c{c}}on.
