Planar Shuffle Product, Co-Addition and the non-associative Exponential
Abstract
Description
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 $Δ$ which leads to a formula for the coefficients of $Δ(f).$
It is also proved that $Δ(EXP) = EXP {\hat \otimes} EXP$ where $EXP$ is the generic planar tree exponential series, see [G]. Systems of quadratic relations for the coefficients of EXP are derived.
12 pages
12 pages