  The Catalan numbers constitute one of the most important sequences in
combinatorics. Catalan objects have been generalized in various directions,
including the classical Fuss-Catalan objects and the rational Catalan
generalization of Armstrong-Rhoades-Williams. We propose a wider generalization
of these families indexed by a composition $s$ which is motivated by the
combinatorics of planar rooted trees; when $s=(2,...,2)$ and $s=(k+1,...,k+1)$
we recover the classical Catalan and Fuss-Catalan combinatorics, respectively.
Furthermore, to each pair $(a,b)$ of relatively prime numbers we can associate
a signature that recovers the combinatorics of rational Catalan objects. We
present explicit bijections between the resulting $s$-Catalan objects, and a
fundamental recurrence that generalizes the fundamental recurrence of the
classical Catalan numbers. Our framework allows us to define signature
generalizations of parking functions which coincide with the generalized
parking functions studied by Pitman-Stanley and Yan, as well as generalizations
of permutations which coincide with the notion of Stirling multipermutations
introduced by Gessel-Stanley. Some of our constructions differ from the ones of
Armstrong-Rhoades-Williams, however as a byproduct of our extension, we obtain
the additional notions of rational permutations and rational trees.
