Some properties of crossings and partitions
type: Item
instance of: scholarly article
title: Some properties of crossings and partitions
zbMATH Open document ID: 0892.05006
DOI: 10.1016/S0012-365X(96)00139-2
author: Philippe Biane
published in: Discrete Mathematics
publication date: 1998-07-19T00:00:00Z
review text: 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.].
Mathematics Subject Classification ID: 05A18
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A30
zbMATH DE Number: 1110010
zbMATH Keywords: noncrossing partitions
zbMATH Keywords: continued fractions
zbMATH Keywords: Motzkin paths
zbMATH Keywords: \(q\)-Stirling numbers
zbMATH Keywords: \(q\)-Bell numbers
reviewed by: C. Krattenthaler
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1110010