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

TL;DR

This paper explores the properties of Raney numbers, providing a recurrence relation, confirming a conjecture about core partitions, and offering a new combinatorial interpretation related to these numbers.

Contribution

It introduces a recurrence relation for Raney numbers, confirms a conjecture on core partitions, and presents a novel combinatorial interpretation of Raney numbers.

Findings

Recurrence relation for Raney numbers derived.

Confirmed Amdeberhan's conjecture on $(s,s+1)$-core partitions.

New combinatorial interpretation of Raney numbers in terms of core partitions.

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 $\lambda$ 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 $\lambda$ with parts that are multiples of $p$.