Square-bounded partitions and catalan numbers
Journal of Algebraic Combinatorics, Tome 34 (2011) no. 1, pp. 1-18.

Voir la notice de l'article dans Electronic Library of Mathematics

Summary: For each integer $k\geq 1$, 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. $\lambda =( \lambda _{1}\geq \cdot \cdot \cdot \geq \lambda _{ k })$ 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.
Mots-clés : keywords partitions, Young diagrams, Catalan numbers, current algebras 
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Bennett, Matthew; Chari, Vyjayanthi; Dolbin, R.J.; Manning, Nathan. Square-bounded partitions and catalan numbers. Journal of Algebraic Combinatorics, Tome 34 (2011) no. 1, pp. 1-18. https://geodesic-test.mathdoc.fr/item/JAC_2011__34_1_a4/