Oriented Sets and Even Permutations: Asymmetric Index Series and Q-Series

One of us introduced the notion of asymmetry index series Γ_F(x₁, x₂,...) of an arbitrary species of structures F in the sense of Joyal. In the context of asymmetric structures the series Γ_F has properties similar to that of classical cycle index series Z_F but is, in general, much more difficult to compute explicitly. Very few closed forms for Γ_F are known. In this paper we obtain closed forms for the asymmetry index series Γ_E±(x₁, x₂,...) and Γ_ALT(x₁, x₂,...) of the species E± and ALT of oriented sets and even permutations. An oriented set is a total order up to an even permutation of its elements. We also obtain the corresponding q-analogues, in the sense of Décoste, arising from the substitutions x_i := (1 - q)ⁱxⁱ/(1 - qⁱ), ⁱ ≥ 1.