Journal of Algebraic Combinatorics

, Volume 34, Issue 1, pp 1–18

Square-bounded partitions and Catalan numbers


  • Matthew Bennett
    • Department of MathematicsUniversity of California
  • Vyjayanthi Chari
    • Department of MathematicsUniversity of California
    • Department of MathematicsUniversity of California
  • Nathan Manning
    • Department of MathematicsUniversity of California
Open AccessArticle

DOI: 10.1007/s10801-010-0260-6

Cite this article as:
Bennett, M., Chari, V., Dolbin, R.J. et al. J Algebr Comb (2011) 34: 1. doi:10.1007/s10801-010-0260-6


For each integer k≥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 λ which is square-bounded, i.e. λ=(λ1≥⋅⋅⋅≥λk) with λ1=k and λk=1, applying the algorithm times gives rise to a set whose cardinality is either the Catalan number ck+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+m variables.


PartitionsYoung diagramsCatalan numbersCurrent algebras
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© The Author(s) 2010