Bijections behind the Ramanujan Polynomials
Bijections behind the Ramanujan Polynomials
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DOI：
10.1006/aama.2001.0738
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发表时间：
2001-07
期刊：
影响因子：
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通讯作者：
William Y. C. Chen;Victor J. W. Guo
中科院分区：
文献类型：
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作者：
William Y. C. Chen;Victor J. W. Guo
拉马努金多项式是由拉马努金在他的研究幂级数反演。在一个关于树数的凯莱公式的方法中，肖尔发现了一个关于不适当边数的精确递归关系，而没有意识到与拉马努金多项式的联系。另一方面，Dumont和Ramamonjisoa独立地对与Ramanujan多项式相关的序列采取了语法方法，并得出了与Shor相同的结论。这是一个巧合，曾意识到，肖尔多项式原来是拉马努金多项式通过一个明确的替代参数。Shor还发现了Ramanujan多项式的一个递归式，它等价于在Zeng的替换下的Bernove-Evans-Wilson递归式，并要求一个组合解释。本文的目的是提出一个双射的Shor递归，或Bernove-Evans-Wilson递归，回答问题的Shor。这样的双射也导致了一个组合的解释递归关系最初由拉马努金。
The Ramanujan polynomials were introduced by Ramanujan in his study of power series inversions. In an approach to the Cayley formula on the number of trees, Shor discovers a refined recurrence relation in terms of the number of improper edges, without realizing the connection to the Ramanujan polynomials. On the other hand, Dumont and Ramamonjisoa independently take the grammatical approach to a sequence associated with the Ramanujan polynomials and have reached the same conclusion as Shor's. It was a coincidence for Zeng to realize that the Shor polynomials turn out to be the Ramanujan polynomials through an explicit substitution of parameters. Shor also discovers a recursion of Ramanujan polynomials which is equivalent to the Berndt-Evans-Wilson recursion under the substitution of Zeng and asks for a combinatorial interpretation. The objective of this paper is to present a bijection for the Shor recursion, or the Berndt-Evans-Wilson recursion, answering the question of Shor. Such a bijection also leads to a combinatorial interpretation of the recurrence relation originally given by Ramanujan.