Séminaire Lotharingien de Combinatoire, B54g (2006), 44 pp.

Eric van Fossen Conrad and Philippe Flajolet

The Fermat Cubic, Elliptic Functions, Continued Fractions, and a Combinatorial Excursion

Abstract. Elliptic functions considered by Dixon in the nineteenth century and related to Fermat's cubic, x3+y3=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 Françon.

Received: July 9, 2005. Accepted: March 2, 2006. Final Version: March 25, 2006.

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