Some properties of crossings and partitions
In the first part of the paper, the author shows that the lattice of noncrossing partitions of \(\{1,2,\dots,n\}\) is isomorphic to a poset constructed naturally from the Cayley graph of the symmetric group \(S_n\) (with the transpositions as generators). In the second part, he derives a continued fraction for the generating function for partitions of \(\{1,2,\dots,n\}\), counted with respect to several statistics, among which is the number of ``restricted crossings'' (see the paper for definition). Specializing, he obtains a continued fraction for the generating function for noncrossing partitions, again counted with respect to several statistics. These results are deduced by suitably modifying a bijection due to \textit{P. Flajolet} [Discrete Math. 32, 125-161 (1980; Zbl 0445.05014)] between partitions and Motzkin paths and applying Flajolet's combinatorial theory of continued fractions [ibid.].
scientific article; zbMATH DE number 1110010