@article {IOPORT.01110088,
    author       = {Randrianarivony, Arthur},
    title        = {$q,p$-Catalan numbers. ($q,p$-analogue des nombres de Catalan.)},
    year         = {1998},
    journal      = {Discrete Mathematics},
    volume       = {178},
    number       = {1-3},
    issn         = {0012-365X},
    pages        = {199-211},
    publisher    = {Elsevier Science B.V. (North-Holland), Amsterdam},
    doi          = {10.1016/S0012-365X(97)81827-4},
    abstract     = {{\it J. F\"urlinger} and {\it J. Hofbauer} [J. Comb. Theory, Ser. A 40, 248-264 (1985; Zbl 0581.05006)], and later the reviewer [Sitzungsber., Abt. II, \"Osterr. 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 imbriqu\'ees". 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 M\'edicis-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.},
    reviewer     = {C.Krattenthaler (Wien)},
    identifier   = {01110088},
}