Squarebounded partitions and Catalan numbers
 Matthew Bennett,
 Vyjayanthi Chari,
 R. J. Dolbin,
 Nathan Manning
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Abstract
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 squarebounded, i.e. λ=(λ _{1}≥⋅⋅⋅≥λ _{ k }) with λ _{1}=k and λ _{ k }=1, applying the algorithm ℓ times gives rise to a set whose cardinality is either the Catalan number c _{ ℓ−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 twoparameter 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 2ℓ+m variables.
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 Title
 Squarebounded partitions and Catalan numbers
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Journal of Algebraic Combinatorics
Volume 34, Issue 1 , pp 118
 Cover Date
 20110801
 DOI
 10.1007/s1080101002606
 Print ISSN
 09259899
 Online ISSN
 15729192
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Partitions
 Young diagrams
 Catalan numbers
 Current algebras
 Authors

 Matthew Bennett ^{(1)}
 Vyjayanthi Chari ^{(1)}
 R. J. Dolbin ^{(1)}
 Nathan Manning ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of California, Riverside, CA, 92521, USA