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.
- 2.
In fact, the same formula gives a pre-Lie algebra for every dendriform algebra.
- 3.
A double corolla is obtained by grafting several copies of the rooted tree with 2 vertices on a common root.
- 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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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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