Trying to work out the amount of permutations in a number set that behaves in this manner:

"This is a reincarnation problem. There are six incarnations of artists. Three are alive and three are waiting in limbo to be reborn. There is always a Child, an Apprentice, and a Old Master. As the Old Master dies the Apprentice takes his place as Master, the Child becomes the Apprentice, and one of the three souls in limbo is reborn as the Child. The Old Master cannot be reborn until his Apprentice has passed away too because after spending all that time together, they would recognize each other and all of their conversations would be super awkward. So, how many different variations of Child, Apprentice, and Old Master are possible? How would you express the equation if there were more than three souls waiting in limbo? Four, five, six, etc.?"

The only formula I remember from my days in math class is:

but I think this is a little too simple to factor in the weird rule about consecutive persons being removed from the pool until they both are "dead". Any thoughts?