The Generating Function of Ternary Trees and Continued Fractions

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dc.creatorGessel, Ira
dc.creatorXin, Guoce
dc.date2005-05-11
dc.date.accessioned2026-07-07T05:19:48Z
dc.date.available2026-07-07T05:19:48Z
dc.descriptionMichael Somos conjectured a relation between Hankel determinants whose entries $\frac 1{2n+1}\binom{3n}n$ count ternary trees and the number of certain plane partitions and alternating sign matrices. Tamm evaluated these determinants by showing that the generating function for these entries has a continued fraction that is a special case of Gauss's continued fraction for a quotient of hypergeometric series. We give a systematic application of the continued fraction method to a number of similar Hankel determinants. We also describe a simple method for transforming determinants using the generating function for their entries. In this way we transform Somos's Hankel determinants to known determinants, and we obtain, up to a power of 3, a Hankel determinant for the number of alternating sign matrices. We obtain a combinatorial proof, in terms of nonintersecting paths, of determinant identities involving the number of ternary trees and more general determinant identities involving the number of $r$-ary trees.
dc.description44 pages, 12 figures
dc.identifierhttps://arxiv.org/abs/math/0505217
dc.identifierhttp://arxiv.org/abs/math/0505217
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/75153
dc.subjectCombinatorics
dc.subject05A15; 05A10, 05A17, 30B70, 33C05
dc.titleThe Generating Function of Ternary Trees and Continued Fractions
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