Generating series of classes of exotic un-ordered configuration spaces

A notion of exotic (ordered) configuration spaces of points on a space $X$ was suggested by Yu.~Baryshnikov. He gave equations for the (exponential) generating series of the Euler characteristics of these spaces. Here we consider un-ordered analogues of these spaces. For $X$ being a complex quasiprojective variety, we give equations for the generating series of classes of these configuration spaces in the Grothendieck ring $K_0({\rm{Var}_{\mathbb{C}}})$ of complex quasiprojective varieties. The answer is formulated in terms of the (natural) power structure over the ring $K_0({\rm{Var}_{\mathbb{C}}})$. This gives equations for the generating series of additive invariants of the configuration spaces such as the Hodge--Deligne polynomial and the Euler characteristic.

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