Catalan Numbers, Their Generalization, and Their Uses

Probably the most prominent among the special integers that arise in combinatorial contexts are the binomial coefficients (~). These have many uses and, often, fascinating interpretations [9]. We would like to stress one particular interpretation in terms of paths on the integral lattice in the coordinate plane, and discuss the celebrated ballot problem using this interpretation. A path is a sequence of points Po P1 9 9 9 Pro, m >I O, where each P, is a lattice point (that is, a point with integer coordinates) and Pz+l, i 1> 0, is obtained by stepping one unit east or one unit north of P,. We say that this is a path from P to Q if Po = P, Pm= Q. It is now easy to count the number of paths.

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