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.
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Acknowledgements
The author is grateful to the referee for his careful reading and for calling our attention to references [3, 15] which consider non-commutative versions of the Rogers–Ramanujan continued fraction. In particular, [15, Theorem 3.3.2] gives an explicit formula for (in our notation) the continued fraction \({\mathscr {A}}(z,q)\) as a quotient of q-series. This gives a novel expression for the generating function \(\frac{{\mathscr {C}}^{(1)}(z,q)}{{\mathscr {C}}^{(1)}(zq,z)}\). These methods could help in finding explicit formulas for \({\mathscr {C}}^{(1)}(z,q)\), and more generally for \({\mathscr {C}}^{(m)}(z,q)\), an interesting problem for future research.
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Méndez, M.A. Shift-plethystic trees and Rogers–Ramanujan identities. Ramanujan J 55, 943–964 (2021). https://doi.org/10.1007/s11139-020-00285-8
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DOI: https://doi.org/10.1007/s11139-020-00285-8