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Shift-plethystic trees and Rogers–Ramanujan identities
The Ramanujan Journal ( IF 0.6 ) Pub Date : 2020-07-26 , DOI: 10.1007/s11139-020-00285-8
Miguel A. Méndez

By studying non-commutative series in an infinite alphabet, we introduce shift-plethystic trees and a class of integer compositions as new combinatorial models for the Rogers–Ramanujan identities. We prove that the language associated to shift-plethystic trees can be expressed as a non-commutative generalization of the Rogers–Ramanujan continued fraction. By specializing the non-commutative series to q-series, we obtain new combinatorial interpretations of the Rogers-Ramanujan identities in terms of signed integer compositions. We introduce the operation of shift-plethysm on non-commutative series and use this to obtain interesting enumerative identities involving compositions and partitions related to Rogers–Ramanujan identities.



中文翻译:

轮状杂种树和罗杰斯-拉马努詹身份

通过研究无限字母中的非可交换级数,我们引入了移位插补树和一类整数组成作为Rogers-Ramanujan身份的新组合模型。我们证明,与移位插补树相关的语言可以表示为Rogers-Ramanujan连续分数的非交换泛化。通过将非交换级数专用于q级数,我们获得了有符号整数组成对Rogers-Ramanujan恒等式的新组合解释。我们介绍了非交换级数上的移位体积运算,并使用它来获得有趣的枚举恒等式,其中涉及与罗杰斯-拉曼努扬恒等式相关的成分和分区。

更新日期:2020-07-26
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