The Raney numbers and -core partitions

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• 作者：Robin D.P. Zhou^a ; ^{dapao2012@163.com" class="auth_mail" title="E-mail the corresponding author} ; Sherry H.F. Yan^b ; ^{huifangyan@hotmail.com" class="auth_mail" title="E-mail the corresponding author}
• 刊名：European Journal of Combinatorics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：59
• 期：Complete
• 页码：114-121
• 全文大小：440 K

文摘

The Raney numbers R_p,r(k)$R_{p,r}\left(k\right)$ 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)$(s,s+1)$-core partitions 74" title="Click to view the MathML source">λ$\lambda$ with parts that are multiples of p$p$. As a corollary, we give a new combinatorial interpretation for the Raney numbers 725ca5c1a" title="Click to view the MathML source">R_p+1,r+1(k)$R_{p+1,r+1}\left(k\right)$ with 0≤r<p$0\le r<p$ in terms of 74ced495008b64fe382d41b2a4c0bc" title="Click to view the MathML source">(kp+r,kp+r+1)$(kp+r,kp+r+1)$-core partitions 74" title="Click to view the MathML source">λ$\lambda$ with parts that are multiples of p$p$.