Free cumulants,Schröder trees,and operads |
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| Institution: | 1. Laboratoire de Mathématiques d''Orsay, Univ. Paris-Sud, CNRS, Université Paris Saclay, Bâtiment 425, 91405 Orsay Cedex, France;2. Laboratoire d''Informatique Gaspard Monge, Université Paris-Est Marne-la-Vallée, 5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France |
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| Abstract: | The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Faà di Bruno algebra, and then to the group of a free operad over Schröder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speicher's formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras. |
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| Keywords: | Free cumulants Noncommutative symmetric functions Hopf algebras Operads |
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