The Reciprocal of 1+ab+aabb+aaabbb+... for NON-COMMUTING a and b, Catalan numbers and non commutative quadratic equations

Arkady Berenstein, Christophe Reutenauer, Vladimir Retakh and Doron Zeilberger

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[Appeared in "Noncommutative Birational Geometry, Representations, and Cluster algebras" (A. Berenstein and V, Retakh, eds.), Contemporary Mathematics volume 592, 2013, Amer. Math. Soc.]
First Posted: June 18, 2012; This version: Oct. 26, 2012.

We all know how to find the reciprocal of a formal power series of commuting variables but things are much more interesting if the variables are (totally!) non-commuting. Then algebra really becomes combinatorics!

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