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On Some Tree-Indexed Series with One and Two Parameters

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Book cover Periods in Quantum Field Theory and Arithmetic (ICMAT-MZV 2014)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 314))

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Abstract

There is a rich algebraic setting involving free pre-Lie algebras and the combinatorics of rooted trees. In this context, one can consider the analog of formal power series, called tree-indexed series. Several interesting such series are known, including one called \(\varOmega \) and its more recent one-parameter and two-parameters generalizations. This survey article explains how one can compute their coefficients using Ehrhart polynomials of lattice polytopes.

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Notes

  1. 1.

    For more on the tree-factorial, see for example [3, 17].

  2. 2.

    In fact, the same formula gives a pre-Lie algebra for every dendriform algebra.

  3. 3.

    A double corolla is obtained by grafting several copies of the rooted tree with 2 vertices on a common root.

  4. 4.

    A linear tree is a rooted tree which is a path graph (no forking vertex), with the root at one end.

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Authors and Affiliations

  1. Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000, Strasbourg, France

    F. Chapoton

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  1. F. Chapoton
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Correspondence to F. Chapoton .

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Editors and Affiliations

  1. Institute of Mathematical Sciences (ICMAT), Spanish National Research Council (CSIC), Madrid, Spain

    José Ignacio Burgos Gil

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kurusch Ebrahimi-Fard

  3. Department of Mathematical Sciences, Durham University, Durham, UK

    Herbert Gangl

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Chapoton, F. (2020). On Some Tree-Indexed Series with One and Two Parameters. In: Burgos Gil, J., Ebrahimi-Fard, K., Gangl, H. (eds) Periods in Quantum Field Theory and Arithmetic. ICMAT-MZV 2014. Springer Proceedings in Mathematics & Statistics, vol 314. Springer, Cham. https://doi.org/10.1007/978-3-030-37031-2_16

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