  We introduce a new class of monomial ideals which we call symmetric shifted
ideals. Symmetric shifted ideals are fixed by the natural action of the
symmetric group and, within the class of monomial ideals fixed by this action,
they can be considered as an analogue of stable monomial ideals within the
class of monomial ideals. We show that a symmetric shifted ideal has linear
quotients and compute its (equivariant) graded Betti numbers. As an application
of this result, we obtain several consequences for graded Betti numbers of
symbolic powers of defining ideals of star configurations.
