Planar Shuffle Product, Co-Addition and the non-associative Exponential

TL;DR

This paper introduces a planar shuffle product for tree polynomials, explores its duality with co-addition, and derives formulas for the exponential series coefficients in the context of planar trees.

Contribution

It defines a new shuffle product for planar trees, establishes its duality with co-addition, and derives quadratic relations for the exponential series coefficients.

Findings

The shuffle product $ox$ is dual to co-addition $ riangle$.

A formula for coefficients of $ riangle(f)$ is provided.

The exponential series satisfies $ riangle(EXP) = EXP oxtimes EXP$.

Abstract

In this note we introduce the concept of a shuffle product $\sq$ for planar tree polynomials and give a formula to compute the planar shuffle product $S \ \sq T$ of two finite planar reduced rooted trees $S, T.$ It is shown that $\sq$ is dual to the co-addition $\Delta$ which leads to a formula for the coefficients of $\Delta(f).$ It is also proved that $\Delta(EXP) = EXP {\hat \otimes} EXP$ where $EXP$ is the generic planar tree exponential series, see [G]. Systems of quadratic relations for the coefficients of EXP are derived.