The Bruhat order on conjugation-invariant sets of involutions in the symmetric group

Let InIn be the set of involutions in the symmetric group SnSn, and for A⊆{0,1,...,n}A⊆{0,1,...,n}, letFnA={σ∈In|σ has a fixed points for some a∈A}. We give a complete characterisation of the sets A   for which FnA, with the order induced by the Bruhat order on SnSn, is a graded poset. In particular, we prove that Fn{1} (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When FnA is graded, we give its rank function. We also give a short new proof of the EL-shellability of Fn{0} (i.e., the set of fixed point-free involutions), which was recently proved by Can, Cherniavsky, and Twelbeck.