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

Lothar Gerritzen
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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 $Δ$ 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.

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Lothar Gerritzen. 2005-02-17. Planar Shuffle Product, Co-Addition and the non-associative Exponential. https://arxiv.org/abs/math/0502378

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