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 square-bounded, i.e. λ=(λ1≥⋅⋅⋅≥λk) with λ1=k and λk=1, applying the algorithm ℓ times gives rise to a set whose cardinality is either the Catalan number cℓ−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 2ℓ+m variables.
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V. Chari was partially supported by the NSF grant DMS-0901253.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Bennett, M., Chari, V., Dolbin, R.J. et al. Square-bounded partitions and Catalan numbers. J Algebr Comb 34, 1–18 (2011). https://doi.org/10.1007/s10801-010-0260-6
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DOI: https://doi.org/10.1007/s10801-010-0260-6