On asymmetric structures

A given structure is said to be asymmetric if its automorphism group reduces to the identity. The problem of enumerating asymmetric structures (and, more generally, to count structures according to stabilizers) is usually solved by making use of Mobius inversion techniques and symmetric functions in the context of group actions. This method of solution was introduced by Rota (1964, 1969) who defined special classes of polynomials which may be called asymmetry indicator polynomials. Subsequent developments following similar ideas can be found in Stockmeyer (1971), White (1975), Rota, Smith and Sagan (1977, 1980), Kerber (1986). We present here another approach to this problem within the theory of species of structures in the sense of Joyal (1981, 1985, 1986). Every species of structures F contains a sub-species F, called the flat part of F, consisting of all asymmetric F-structures. We introduce an asymmetry indicator series ΓF(x1, x2, x3,…) by means of which we study the correspondence F↦F in connection with the various operations existing in the theory of species of structures. The main result is that the ΓF behaves with respect to the combinatorial operations of sum, product, substitution and differentiation as does the classical cycle indicator series ZF. As a consequence, the asymmetry indicator series can be applied to the systematic classification and enumeration of asymmetric F-structures when the species F is defined (explicitly or recursively) by combinatorial equations. We illustrate the method on particular species (including enriched trees and rooted trees) and a table of ΓF is given for the atomic species concentrated on small cardinalities. Examples show that ΓF contains information independent of that in ZF.