Universal Singular Exponents in Catalytic Variable Equations

Catalytic equations appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that the asymptotic estimate for the coefficients of the solutions of (so-called) positive catalytic equations has a universal asymptotic behavior. In particular, this provides a rationale why the number of maps of size n in various planar map classes grows asymptotically like \(c\cdot n^{-5/2} \gamma ^n\), for suitable positive constants c and \(\gamma \). Essentially we have to distinguish between linear catalytic equations (where the subexponential growth is \(n^{-3/2}\)) and non-linear catalytic equations (where we have \(n^{-5/2}\) as in planar maps). The Proofs are based on a delicate analysis of systems of polynomials equations and singularity analysis and are omitted for lack of space. positive. (Supported by the Ministerio de Economia y Competitividad grant MTM2017-82166-P, and by the Special Research Program F50 Algorithmic and Enumerative Combinatorics of the Austrian Science Fund. Research supported by the Austrian Science Foundation FWF, project F 50-02.).