Pivot swap permutations in a 4 ball break
Basic 4 ball break
In a 4 ball break the first pioneer (P1) and forward pioneer (Pf) swap while the pivot (Pi) stays the same
4 ball break
Shot  
1  Roquet P1 
2  Hoop approach. Striker ball position critical. Reception ball less so 
3  Run hoop 
4  Roquet reception ball 
5  Croquet reception ball to new Pf. Striker ball within range of Pi 
6  Roquet pivot 
7  Take off to old Pf ≡ next P1 
Pivot swap
Start with roquet of P1.
P1 is rushed to near Pi after it becomes the reception ball.
It must have been accurately placed in a different position than for a normal 4 ball break hoop,
so you need to know 2 hoops in advance that a pivot swap is required.
Pivot swap

From hoop 1

From hoop 2

From hoop 3

From hoop 4

From hoop 5

From hoop 6
The sequence of shots from nback is a reflection in the WE centre line of that from n for n ∈ [1,4].
The last shot of a pivot swap is the same as the last shot of a 3 ball break.
Pivot swaps at even numbered hoops are much harder than those at odd numbered hoops.
Arbitrary P1, Pf and Pi permutations
The objective is to have the required balls at P1, Pf and Pi at, say, 2 back.
Start at hoop 1 with balls B1, B2 and B3 at P1, Pi and Pf respectively.
Hoop  Move  P1  Pi  Pf 
1  Normal  B1  B2  B3 
2  Normal  B3  B2  B1 
…

2b  Normal  B3  B2  B1 
Hoop  Move  P1  Pi  Pf 
1  P swap  B1  B2  B3 
2  Normal  B3  B1  B2 
…

2b  Normal  B3  B1  B2 
Hoop  Move  P1  Pi  Pf 
1  Normal  B1  B2  B3 
2  P swap  B3  B2  B1 
3  Normal  B1  B3  B2 
…

1b  Normal  B1  B3  B2 
2b  Normal  B2  B3  B1 
Hoop  Move  P1  Pi  Pf 
1  Normal  B1  B2  B3 
2  P swap  B3  B2  B1 
3  P swap  B1  B3  B2 
4  Normal  B2  B1  B3 
…

2b  Normal  B2  B1  B3 
Hoop  Move  P1  Pi  Pf 
1  P swap  B1  B2  B3 
2  P swap  B3  B1  B2 
3  Normal  B2  B3  B1 
…

1b  Normal  B2  B3  B1 
2b  Normal  B1  B3  B2 
Hoop  Move  P1  Pi  Pf 
1  P swap  B1  B2  B3 
2  P swap  B3  B1  B2 
3  P swap  B2  B3  B1 
4  Normal  B1  B2  B3 
…

2b  Normal  B1  B2  B3 
Example 1
Striking ball R arrives at hoop 5 with Y as P1, K as Pi and B as Pf.
We want to reach 2 back with B as P1, Y as Pi and K as Pf to make a 2 hoop leave after running 3 back.
This set of moves is required:
Hoop  Move  P1  Pi  Pf 
5  P swap  Y  K  B 
6  Normal  B  Y  K 
1b  Normal  K  Y  B 
2b   B  Y  K 
Example 1
Croquet home