Counting asymmetric enriched trees

Abstract

A structure is said to be asymmetric if its automorphism group reduces to the identity. We present new recursive and explicit formulas for the enumeration of asymmetric R-enriched trees and R-enriched rooted trees where R is an arbitrary given species of structures. Depending on the choice of R these tree-like structures include: ordinary, planar, topological, oriented, 1-2-3, cyclic, and permutation (rooted and unrooted) trees. The problem of counting asymmetric R-enriched trees is transformed into a problem of symbolic computation. Our basic tools are the asymmetry index series of a species and a generalization of the Otter-Robinson-Leroux formula consisting in an isomorphism relating the species of enriched trees and enriched rooted trees. We give examples and tables using computer algebra.