Planar Binomial Coefficients

Abstract

The notion of binomial coefficients $\left(\genfrac{}{}{0ex}{}{T}{S}\right)$ of finite planar, reduced rooted trees $T,S$ is defined and a recursive formula for its computation is shown. The nonassociative binomial formula $$(1+x{)}^T=\sum _S(\genfrac{}{}{0ex}{}{T}{S})x^S$$ for powers relative to $T$ is derived. Similarly binomial coefficients $\left(\genfrac{}{}{0ex}{}{T}{S,V}\right)$ of the second kind are introduced and it is shown that $(x\otimes 1+1\otimes x{)}^T=\sum _{S,V}(\genfrac{}{}{0ex}{}{T}{S,V})(x^S\otimes x^V)$ The roots $\sqrt[T]{1+x}=(1+x{)}^{T^{-1}}$ which are planar power series $f$ such that $f^T=1+x$ are considered. Formulas for their coefficients are given.

8 pages