Even permutations and oriented sets: their shifted asymmetry index series

The present authors introduced recently (see [Adv. in Appl. Math. 32 (2004) 576]) the notion of shifted asymmetry index series denoted , for an arbitrary species of structures, F, and a series of weights, ξ. This series generalizes the classical asymmetry index series, , of G. Labelle [Discrete Math. 99 (1992) 141] to take into account the substitution of species with non-zero constant term [Combinatoire Énumérative, Lecture Notes in Math., vol. 1234, Springer-Verlag, 1986, pp. 126–159]. In [Adv. in Appl. Math. 32 (2004) 576], we can find explicit formulas for the shifted asymmetry index series of usual species (sets, cycles, permutation, trees, …). The goal of this paper is to complement [Adv. in Appl. Math. 32 (2004) 576] by computing closed formulas for the series and , of the species of oriented sets and ALT of even permutation of their elements. Oriented sets are, by definition, totally ordered sets modulo an even permutation of their elements. Such structures were used, for example, by Pólya and Read [Combinatorial Enumeration of Groups, Graphs and Chemical Compounds, Springer-Verlag, 1987] in the context of combinatorial chemistry problems (vertex-sets of oriented simplices). [Copyright &y& Elsevier]