Shift-plethysm, Hydra continued fractions, and m-distinct partitions

TL;DR

This paper introduces hydra continued fractions, generalizes Rogers-Ramanujan fractions, and connects them to m-distinct partitions and compositions, providing new combinatorial generating functions and interpretations.

Contribution

It presents hydra continued fractions as a new generalization, with combinatorial interpretations and links to partition and composition generating functions.

Findings

Hydra continued fractions generalize Rogers-Ramanujan fractions.

New generating functions for compositions and partitions are derived.

Connections between continued fractions, partitions, and compositions are established.

Abstract

We introduce the hydra continued fractions, as a generalization of the Rogers-Ramanujan continued fractions, and give a combinatorial interpretation in terms of shift-plethystic trees. We then show it is possible to express them as a quotient of m-distinct partition generating functions, and in its dual form as a quotient of the generating functions of compositions with contiguous rises upper bounded by m-1. We obtain new generating functions for compositions according to their local minima, for partitions with a prescribed set of rises, and for compositions with prescribed sets of contiguous differences.