Combinatorial interpretations and q-analogues of the catalan numbers

The Catalan numbers form one of the more frequently encountered counting sequences in combinatorics. In this thesis, the Catalan numbers are developed in the context of their roots in two historical problems; the problem of determining the number of ways in which an n-gon can be divided into triangular regions by means of non-intersecting diagonals, and the problem of determining the number of ways in which parentheses can be inserted into a product of n factors. Several structures which are counted by the Catalan sequence are then discussed, including sub-diagonal lattice paths and trivalent trees. Finally, two q-analogues of the Catalan numbers are presented.