Combinatoric Results for Graphical Enumeration and the Higher Catalan Numbers

We summarize some combinatoric problems solved by the higher Catalan numbers. These problems are generalizations of the combinatoric problems solved by the Catalan numbers. The generating function of the higher Catalan numbers appeared recently as an auxiliary function in enumerating maps and explicit computations of the asymptotic expansion of the partition function of random matrices in the unitary ensemble case. We give combinatoric proofs of the formulas for the number of genus 0 and genus 1 maps. 1. Higher Catalan Numbers The Catalan numbers solve a number of classical combinatoric problems such as the “Euler Polygon Division Problem”: how many ways are there to divide a marked polygon with j + 2 sides into triangles using edges and diagonals [3, 7, 8, 12, 16] (see figure 1). Figure 1. A polygon with 4 + 2 = 6 sides divided into 4 triangles using edges and diagonals They count the number of right-left paths along a 1-Dimensional integer lattice which stay to the right of 0 and go from 0 to 0 in 2j steps; equivalently they count Dyck paths from (0, 0) to (2j, 0) [1, 4, 15, 18]. They count the number of ways for 2j customers to line up, with j customers having only a 2-dollar bill and j customers having only a 1-dollar bill, to purchase 1-dollar widgets, so that each customer receives exact change. They count the number of non-crossing handshakes possible across a round table between n people [5]. In this paper we will explore a generalization of the Catalan numbers, the higher Catalan numbers. We will show that this generalization solves enumerative problems that are natural generalizations of the problems solved by the Catalan numbers. We will then highlight their appearance in recent results on map enumeration problems. Let (1.1) z(s) = 1 + ∞