Geodesic
The # product in combinatorial Hopf algebras
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (2011).

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We show that the # product of binary trees introduced by Aval and Viennot (2008) is in fact defined at the level of the free associative algebra, and can be extended to most of the classical combinatorial Hopf algebras. 
@article{DMTCS_2011_special_260_a5,
     author = {Aval, Jean-Christophe and Novelli, Jean-Christophe and Thibon, Jean-Yves},
     title = {The # product in combinatorial {Hopf} algebras},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)},
     year = {2011},
     doi = {10.46298/dmtcs.2892},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2892/}
}
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%A Thibon, Jean-Yves
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%V DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)
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Aval, Jean-Christophe; Novelli, Jean-Christophe; Thibon, Jean-Yves. The # product in combinatorial Hopf algebras. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (2011). doi : 10.46298/dmtcs.2892. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2892/

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