Square Partitions and Catalan Numbers

For each integer $k\ge 1$, we define an algorithm which associates to a partition whose maximal value is at most $k$ a certain subset of all partitions. In the case when we begin with a partition $\lambda$ which is square, i.e $\lambda ={\lambda }_1\ge \mathrm{.}\mathrm{.}\mathrm{.}\ge {\lambda }_k>0$, and ${\lambda }_1=k,{\lambda }_k=1$, then applying the algorithm $\mathrm{\ell }$ times gives rise to a set whose cardinality is either the Catalan number $c_{\mathrm{\ell }-k+1}$ (the self dual case) or twice the Catalan number. The algorithm defines a tree and we study the propagation of the tree, which is not in the isomorphism class of the usual Catalan tree. The algorithm can also be modified to produce a two–parameter family of sets and the resulting cardinalities of the sets are the ballot numbers. Finally, we give a conjecture on the rank of a particular module for the ring of symmetric functions in $2\mathrm{\ell }+m$ variables.