Partitions with \(k\) crossings
The author proves that the ordinary generating function \(S_k(x)\) for the number \(S_k(n)\) of partitions of the set \(\{1,2,\dots,n\}\) with \(k\) crossings [see, e.g., \textit{R. Simion}, J. Comb. Theory, Ser. A 66, No. 2, 270-301 (1994; Zbl 0803.05003)] is a rational function of \(x\) and the ordinary generating function of the Catalan numbers. In particular, \(S_1(n)= \left(\begin{smallmatrix} 2n-5\\ n-4\end{smallmatrix}\right)\). This sequence first found by A. Cayley in 1890 is used in various counting problems.
scientific article; zbMATH DE number 1347444
scientific article