\(q,p\)-Catalan numbers
\textit{J. Fürlinger} and \textit{J. Hofbauer} [J. Comb. Theory, Ser. A 40, 248-264 (1985; Zbl 0581.05006)], and later the reviewer [Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 198, No. 4-7, 171-199 (1989; Zbl 0722.05012)], studied trivariate extensions of Catalan numbers, denoted \(C_n(x,a,b)\), which contain most of the various extensions of Catalan numbers that are scattered in the literature as special cases. Roughly speaking, \(C_n(x,a,b)\) counts Catalan paths with respect to three statistics which depend on the turns of the paths. In the paper under review, new surprising combinatorial interpretations of the numbers \(C_n(q,p,p^{-1})\) are given. What the author shows is that \(C_n(q,p,p^{-1})\) is equal to certain generating functions for permutations without crossings, respectively without ``paires imbriquées''. He provides several interesting specializations as well. The proofs rest on Flajolet's combinatorial theory [Discrete Math. 32, 125-161 (1980; Zbl 0445.05014)] of continued fractions and on a variation of the Foata-Zeilberger-de Médicis-Viennot bijection [Stud. Appl. Math. 83, No. 1, 31-59 (1990; Zbl 0738.05001); Adv. Appl. Math. 15, No. 3, 262-304 (1994; Zbl 0812.05074)] between permutations and labelled Dyck paths.
scientific article; zbMATH DE number 1110088