  There are (at least) two different approaches to define equivariant analogue
of the Euler charateristic for a space with a finite group action. The first
one defines it as an element of the Burnside ring of the group. The second
approach emerged from physics and includes the orbifold Euler characteristic
and its higher order versions. Here we suggest a way to merge the two
approaches together defining (in a certain setting) higher order Euler
characteristics with values in the Burnside ring of a group. We give Macdonald
type equations for these invariants. We also offer generalized ("motivic")
versions of these invariants and formulate Macdonald type equations for them as
well.
