The noncommutative geode

TL;DR

The paper extends the Lagrange-series framework to symmetric and noncommutative settings by introducing the geode $\gamma$ and its noncommutative counterpart, enabling detailed combinatorial interpretations tied to parking functions, plane trees, and $0$-Hecke algebra representations. It provides explicit constructions via $S_n^{-1}$ operators, explores expansions in multiple bases, and presents noncommutative Gessel-like factorisations, while introducing $k$-geodes and $e$-Lagrange variants connected to Schröder trees. The work offers closed-form expressions, generating functions, and extensive tables for low-degree cases, establishing a unified algebraic-combinatorial framework for enumerating parking-structure–related objects under Lagrange transforms. These results deepen the link between noncommutative symmetric functions, parking function representations, and classical combinatorial families such as Catalan and Schröder objects, with potential applications in algebraic combinatorics and representation theory.

Abstract

We investigate the geode and some of its generalizations from the point of view on noncommutative symmetric functions.