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Some properties of crossings and partitions. (English) Zbl 0892.05006
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 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.].

MSC:
05A18 Partitions of sets
05A05 Permutations, words, matrices
05A15 Exact enumeration problems, generating functions
05A30 \(q\)-calculus and related topics
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