A GARDEN OF k-CATALAN STRUCTURES

The aim in this paper is to collect in one place a list of currently known and new structures enumerated by the k-ary numbers. Some of the structures listed already exist in the folk-lore, especially those that are easy generalizations of known combinatorial structures enumerated by the Catalan numbers. We will provide outlines on how the proofs for the Catalan structures generalize, and give proofs when the Catalan proof does not easily generalize. In the process, we define a new generalization of Young tableaux, which allow for sets of values in the cells of the tableaux. In addition, we investigate in more detail one of the new k-Catalan structures, namely a tiling of the staircase An in the R 2 plane. For the latter, we exhibit a bijection to k-ary trees and k-ary paths and use this bijection to enumerate statistics on the staircase tilings.