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arXiv · 1512.08080

The Raney Numbers and $(s,s+1)$-Core Partitions

Robin DaPao Zhou
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Abstract

The Raney numbers $R_{p,r}(k)$ are a two-parameter generalization of the Catalan numbers. In this paper, we obtain a recurrence relation for the Raney numbers which is a generalization of the recurrence relation for the Catalan numbers. 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 $p$. We then give a new combinatorial interpretation for the Raney numbers $R_{p+1,r+1}(k)$ with $0\leq r<p$ in terms of $(kp+r,kp+r+1)$-core partitions $λ$ with parts that are multiples of $p$.

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Robin DaPao Zhou. 2015-12-26. The Raney Numbers and $(s,s+1)$-Core Partitions. https://arxiv.org/abs/1512.08080

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