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- W2963424014 abstract "The Raney numbers $R_{p,r}(n)$ are a two-parameter generalization of the Catalan numbers that were introduced by Raney in his investigation of functional composition patterns cite{Raney}. We give a new combinatorial interpretation for all Raney numbers in terms of planar embeddings of certain collections of trees, a construction that recovers the usual interpretation of the $p$-Catalan numbers in terms of $p$-ary trees via the specialization $R_{p,1}(n) =_{p} c_n$. Our technique leads to several combinatorial identities involving the Raney numbers and ordered partitions. We then give additional combinatorial interpretations of specific Raney numbers, including an identification of $R_{p^2,p}(n)$ with oriented trees whose vertices satisfy the source or sink property. We close with comments applying these results to the enumeration of connected (non-elliptic) $A_2$ webs that lack an internal cycle." @default.
- W2963424014 created "2019-07-30" @default.
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- W2963424014 date "2015-01-01" @default.
- W2963424014 modified "2023-09-26" @default.
- W2963424014 title "Generalized Catalan numbers and the enumeration of planar embeddings." @default.
- W2963424014 hasPublicationYear "2015" @default.
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