×

The generating function of ternary trees and continued fractions. (English) Zbl 1098.05006

Summary: Michael Somos conjectured a relation between Hankel determinants whose entries \({1\over 2n+1}{3n\choose n}\) 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.

MSC:

05A15 Exact enumeration problems, generating functions
05A10 Factorials, binomial coefficients, combinatorial functions
05A17 Combinatorial aspects of partitions of integers
30B70 Continued fractions; complex-analytic aspects
33C05 Classical hypergeometric functions, \({}_2F_1\)
PDFBibTeX XMLCite
Full Text: arXiv EuDML EMIS