element which separately acts on { 1   . . .   p } and { p + 2   . . .   n } as required  we have that w $^{′}$u = w . As an example  suppose that p = q = 3  and let u be the signed permutation 316254. To unscramble the 312  we must multiply on the left by 1 ↦→ 2  2 ↦→ 3  3 ↦→ 1  and to unscramble the 65 we must multiply on the left by 5 ↦→ 6   6 ↦→ 5. Thus we multiply u on the left by w ′ = 231465 to get w $^{′}$u = w = 125364. Note that a permutation w having the properties above is completely determined by the positions (in the one-line notation) of 1   . . .   p among the first n − 1 spots  which can be chosen freely. Thus there are ( n − 1 p ) such w   and hence ( n − 1 p ) closed ˜ K -orbits  as claimed. Definition 5.3.2. Let Q ∈ ˜ K \ X be a closed orbit. Call the flag wB ∈ Q   where w has the properties listed in the proof of Proposition 5.3.1  the standard representative of Q . For w ∈ W such that wB is the standard representative of some closed orbit Q   define I$_{w}$ := { i ∈ { 1   . . .   n − 1 } | w ( i ) > p + 1 } . For each i ∈ I$_{w}$   define Finally  define C ( i ) := # { j | i < j ≤ n − 1   w ( j ) ≤ p } . f ( w ) := ∑ i ∈ I$_{w}$ C ( i ) . Then we have the following formula for the S -equivariant class of the closed orbit Q : Proposition 5.3.3. Let Q = ˜ K · wB be any closed orbit  with wB the standard representative. 124