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arXiv · 2004.05503

Shift-Plethystic Trees and Rogers-Ramanujan Identitites

Miguel A. Mendez
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Abstract

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 noncommutative series to $q$-series we obtain new combinatorial interpretations to 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.

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Miguel A. Mendez. 2020-04-11. Shift-Plethystic Trees and Rogers-Ramanujan Identitites. https://arxiv.org/abs/2004.05503

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