Biane, Philippe Some properties of crossings and partitions. (English) Zbl 0892.05006 Discrete Math. 175, No. 1-3, 41-53 (1997). 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.]. Reviewer: C.Krattenthaler (Wien) Cited in 2 ReviewsCited in 74 Documents MSC: 05A18 Partitions of sets 05A05 Permutations, words, matrices 05A15 Exact enumeration problems, generating functions 05A30 \(q\)-calculus and related topics Keywords:noncrossing partitions; continued fractions; Motzkin paths; \(q\)-Stirling numbers; \(q\)-Bell numbers Citations:Zbl 0445.05014 PDFBibTeX XMLCite \textit{P. Biane}, Discrete Math. 175, No. 1--3, 41--53 (1997; Zbl 0892.05006) Full Text: DOI References: [1] Biane, P., Permutations suivant le type de biexcédance et interprétation d’une fraction continue de Heine, Eur. J. Combin., 14, 277-284 (1993) · Zbl 0784.05005 [2] Biane, P., Permutation model for semi-circular systems and quantum random walks, Pacific. J. Math., 171, 373-387 (1995) · Zbl 0854.60070 [3] Bozejko, M.; Speicher, R., An example of a generalized brownian motion, Comm. Math. Phys., 137, 519-531 (1991) · Zbl 0722.60033 [4] Edelman, P. H., Chain enumeration and non-crossing partitions, Discrete Math., 31, 171-180 (1980) · Zbl 0443.05011 [5] Edelman, P. H., Multichains, non-crossing partitions and trees, Discrete Math., 40, 171-179 (1982) · Zbl 0496.05007 [6] Flajolet, P., Combinatorial aspects of continued fractions, Discrete Math., 32, 125-161 (1980) · Zbl 0445.05014 [7] Kreweras, G., Sur les partitions non-croisées d’un cycle, Discrete Math., 1, 333-350 (1972) · Zbl 0231.05014 [8] Nica, A., R-transforms of free joint distributions, and non-crossing partitions, J. Funct. Anal., 135, 271-296 (1996) · Zbl 0837.60008 [9] Simion, R.; Ullman, D., On the structure of the lattice of non-crossing partitions, Discrete Math., 98, 193-206 (1991) · Zbl 0760.05004 [10] Speicher, R., A new example of “independance” and “white noise”, Probab. Theory Rel. Fields, 84, 141-159 (1990) · Zbl 0671.60109 [11] Speicher, R., Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Annalen, 298, 611-628 (1994) · Zbl 0791.06010 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.