Polygon dissections and Euler, Fuss, Kirkman and Cayley numbers
  We give a short proof for a formula for the number of divisions of a convex (sn+2)-gon along non-crossing diagonals into (sj+2)-gons, where 1<=j<=n-1. In other words, we consider dissections of an (sn+2)-gon into pieces which can be further subdivided into (s+2)-gons. This formula generalizes the formulas for classical numbers of polygon dissections: Euler-Catalan number, Fuss number and Kirkman-Cayley number.   Our proof is elementary and does not use the method of generating functions. 
Jozef H. Przytycki, Adam S. Sikora