2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75771Elliptic 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 Pólya 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{ç}on.44 pages; submitted to "Seminaire Lotharingien de Combinatoire" (journal), July 2005CombinatoricsProbability05A15; 30B70; 33C75; 60C05text