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.

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。