The Raney numbers and [formula omitted]-core partitions

Abstract

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<p in terms of (kp+r,kp+r+1)-core partitions λ with parts that are multiples of p.