 
Summary: Combinatorics of the free Baxter algebra
Marcelo Aguiar
Department of Mathematics
Texas A&M University, College Station, TX, USA
maguiar@math.tamu.edu
Walter Moreira
Department of Mathematics
Texas A&M University, College Station, TX, USA
wmoreira@math.tamu.edu
Submitted: October 7, 2005; Accepted: February 9, 2006
Mathematics Subject Classification: 05A15, 08B20, 16W99
Abstract
We study the free (associative, noncommutative) Baxter algebra on one gen
erator. The first explicit description of this object is due to EbrahimiFard and
Guo. We provide an alternative description in terms of a certain class of trees,
which form a linear basis for this algebra. We use this to treat other related cases,
particularly that in which the Baxter map is required to be quasiidempotent, in a
unified manner. Each case corresponds to a different class of trees.
Our main focus is on the underlying combinatorics. In several cases, we pro
vide bijections between our various classes of trees and more familiar combinatorial
