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Báo cáo toán học: "Combinatorics of the free Baxter algebra"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Combinatorics of the free Baxter algebra. | 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 Oct 7 2005 Accepted Feb 9 2006 Published Feb 22 2006 Mathematics Subject Classification 05A15 08B20 16W99 Abstract 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 Schroder 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 Today 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. Both authors supported in part by NSF grant DMS-0302423. We thank Kurusch Ebrahimi-Fard for an explanation of the paper 6 which led us to the results of this paper. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 .