Square-bounded partitions and catalan numbers

Bennett, Matthew ; Chari, Vyjayanthi ; Dolbin, R.J. ; Manning, Nathan

Journal of Algebraic Combinatorics, Tome 34 (2011) no. 1, pp. 1-18.

Résumé

Summary: 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-bounded, i.e. $\lambda =({\lambda }_1\ge \cdot \cdot \cdot \ge {\lambda }_k)$ with ${\lambda }_1=k$ and ${\lambda }_k=1$, applying the algorithm $\ell$ times gives rise to a set whose cardinality is either the Catalan number $c_{\ell -k+1}$ (the self dual case) or twice that 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 $2\ell +m$ variables.

Détail

Mots-clés : keywords partitions, Young diagrams, Catalan numbers, current algebras