# Some Fine Catalan Identities

## ABSTRACT

The study of the sequence of Catalan numbers $C_{n}$, given by

$$C_{n} = {1 \over (n+1)}bin(2n,n), n \geq 0,$$

seems almost to have gained a life of its own; Richard Stanley, in his Enumerative Combinatorics II, gives an exercise in 66 parts, each part giving a different structure enumerated by this sequence according to some parameter of interest. The sequence of Fine numbers $F_{n}$ is perhaps less well-known, but is linked to the Catalan numbers by the equation $$C_{n} = 2F_{n} + F_{n-1}, n \geq 1; F_{0} = 1,$$ although this is not how the sequence might be defined in the first instance, and there are other ways in which the sequences are intertwined.

But to verfify this equation it would be very nice to have a partition of the Catalan structures into Fine structures in which the equation is reflected in the subsets of the partition. This talk takes up the quest for such a demonstration, giving a generally accessible guided research tour that culminates in a successful proof technique for this identity amongst others.

Speaker's Contact Info: drogers(at-sign)math.hawaii.edu