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

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.

12 pages

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