Arbres, arborescences et racines carrées symétriques
✍ Scribed by Pierre Bouchard; Yves Chiricota; Gilbert Labelle
- Publisher
- Elsevier Science
- Year
- 1995
- Tongue
- English
- Weight
- 425 KB
- Volume
- 139
- Category
- Article
- ISSN
- 0012-365X
No coin nor oath required. For personal study only.
✦ Synopsis
Let C be the field of complex numbers and d be the set of atomic species (up to isomor-
if the equation Gz = F holds. Similarly, a species G is said to be a symmetric square root of a species F if E2(G) = F holds, where E, denotes the species of unordered pairs. Although not every species possesses a square root, we prove that it always possesses at least one (and at most two) symmetric square roots. In particular, we show that the species X of singletons has a unique symmetric square root whose expansion begins with the terms -I -X-E,(X)+ X2 + XE,(X)-X3 + ... We also show that, up to an affine transformation, the species of rooted trees is one of the two symmetric square roots of the species of frees. In this case, the other symmetric square root has rational coefficients and its combinatorial interpretation is unknown. We conclude with some generalizations and directions for future investigations. Dhfinition 1.1. Soit F'EC[[&]]. On dira que G est une twine carrke symdrique de F si l'itquation suivante est vitrifike: Ez(G)=F.