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
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
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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