The Raney numbers and (s,s+1)-core partitions

The Raney numbers Rp,r(k) are a two-parameter generalization of the Catalan numbers. In this paper, we give a combinatorial proof for a recurrence relation of the Raney numbers in terms of coral diagrams. Using this recurrence relation, we confirm a conjecture posed by Amdeberhan concerning the enum...

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Bibliographic Details
Published in:European journal of combinatorics 2017-01, Vol.59, p.114-121
Main Authors: Zhou, Robin D.P., Yan, Sherry H.F.
Format: Article
Language:English
Subjects:
Combinatorial analysis
Corals
Enumeration
Partitions
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Summary:The Raney numbers Rp,r(k) are a two-parameter generalization of the Catalan numbers. In this paper, we give a combinatorial proof for a recurrence relation of the Raney numbers in terms of coral diagrams. Using this recurrence relation, we confirm a conjecture posed by Amdeberhan concerning the enumeration of (s,s+1)-core partitions λ with parts that are multiples of p. As a corollary, we give a new combinatorial interpretation for the Raney numbers Rp+1,r+1(k) with 0≤r
ISSN:0195-6698
1095-9971
DOI:10.1016/j.ejc.2016.08.003