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{\bf Marcelo Aguiar and Walter Moreira}
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{\bf Combinatorics of the Free Baxter Algebra}
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We study the free (associative, non-commutative) Baxter algebra on one
generator. The first explicit description of this object is due to
Ebrahimi-Fard 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 quasi-idempotent, 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 provide bijections between our various classes of trees and more
familiar combinatorial objects including certain Schr\"oder paths and
Motzkin paths.  We calculate the dimensions of the homogeneous
components of these algebras (with respect to a bidegree related to
the number of nodes and the number of angles in the trees) and the
corresponding generating series. An important feature is that the
combinatorics is captured by the idempotent case; the others are
obtained from this case by various binomial transforms.  We also
relate free Baxter algebras to Loday's dendriform trialgebras and
dialgebras. We show that the free dendriform trialgebra (respectively,
dialgebra) on one generator embeds in the free Baxter algebra with a
quasi-idempotent map (respectively, with a quasi-idempotent map and an
idempotent generator). This refines results of Ebrahimi-Fard and Guo.

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