The Raney numbers and -core partitions

详细信息    查看全文

• 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 16300622&_mathId=si5.gif&_user=111111111&_pii=S0195669816300622&_rdoc=1&_issn=01956698&md5=07facba986d22683d73dbf1761348376" title="Click to view the MathML source">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 16300622&_mathId=si4.gif&_user=111111111&_pii=S0195669816300622&_rdoc=1&_issn=01956698&md5=3efbb010c82be2ab8ee122409760231b" title="Click to view the MathML source">(s,s+1)$(s,s+1)$-core partitions 16300622&_mathId=si7.gif&_user=111111111&_pii=S0195669816300622&_rdoc=1&_issn=01956698&md5=cd522a1a9f78d0bf30cf035ed50af174" title="Click to view the MathML source">λ$\lambda$ with parts that are multiples of 16300622&_mathId=si8.gif&_user=111111111&_pii=S0195669816300622&_rdoc=1&_issn=01956698&md5=f0641c1697a63c9baf5a516e86d46055" title="Click to view the MathML source">p$p$. As a corollary, we give a new combinatorial interpretation for the Raney numbers 16300622&_mathId=si9.gif&_user=111111111&_pii=S0195669816300622&_rdoc=1&_issn=01956698&md5=f16e98630e7ca60bb6d1607725ca5c1a" title="Click to view the MathML source">R_p+1,r+1(k)$R_{p+1,r+1}\left(k\right)$ with 16300622&_mathId=si10.gif&_user=111111111&_pii=S0195669816300622&_rdoc=1&_issn=01956698&md5=fb2f2354304d524ff885d4d623455b96" title="Click to view the MathML source">0≤r<p$0\le r<p$ in terms of 16300622&_mathId=si11.gif&_user=111111111&_pii=S0195669816300622&_rdoc=1&_issn=01956698&md5=c174ced495008b64fe382d41b2a4c0bc" title="Click to view the MathML source">(kp+r,kp+r+1)$(kp+r,kp+r+1)$-core partitions 16300622&_mathId=si7.gif&_user=111111111&_pii=S0195669816300622&_rdoc=1&_issn=01956698&md5=cd522a1a9f78d0bf30cf035ed50af174" title="Click to view the MathML source">λ$\lambda$ with parts that are multiples of 16300622&_mathId=si8.gif&_user=111111111&_pii=S0195669816300622&_rdoc=1&_issn=01956698&md5=f0641c1697a63c9baf5a516e86d46055" title="Click to view the MathML source">p$p$.