q-Catalan numbers

Abstractq-analogs of the Catalan numbers Cn = (1(n + 1))(n2n) are studied from the view-point of Lagrange inversion. The first, due to Carlitz, corresponds to the Andrews-Gessel-Garsia q-Lagrange inversion theory, satisfies a nice recurrence relation and counts inversions of Catalan words. The second, tracing back to Mac Mahon, arise from Krattenthaler's and Gessel and Stanton's q-Lagrange inversion formula, have a nice explicit formula and enumerate the major index. Finally a joint generalization is given which includes also the Polya-Gessel q-Catalan numbers