Shift-plethystic trees and Rogers–Ramanujan identities

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 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.