Arbres, arborescences et racines carrées symétriques

Abstract

Let C be the field of complex numbers and A be the set of atomic species (up to isomorphism). A species G in C [[ A ]] is said to be a square root of a species F in C [[ A ]] if the equation G 2 = F holds. Similarly, a species G is said to be a symmetric square root of a species F if E 2( G) = F holds, where E 2 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 − 1 − X − E 2( X) + X 2 + X E 2( X) − X 3 + ⋯. 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 trees. 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.