I have an ordered set, with three subsets; S = (A={1, 5}, B={3, 6}, C={8}). I currently want to figure out how many permutations there are if each permutation step is restricted by subset A or B swapping an element with subset C. While I can figure out this with bruteforce, I came here to get an understanding how a mathematician would tackle this and what is this type of permutation actually called?
Context: I work on a game called Onitama, a board game, where each player has two move cards (subset A and B) and an idle card (subset C). For each turn the player must swap one of their cards with the idle card. A player is free to pick which of their own cards to swap, hence the unordered subsets. I wanted to use permutations to figure out how many turns could be played until the cards returned to their original subsets.
Example steps (of how I assume this would work):
- S = (A={1,8}, B={3,6}, C={5})
- S = (A={1,8}, B={3,5}, C={6})
S = (A={6,8}, B={3,5}, C={1})
...
N. S = (A={1,5}, B={3,6}, C={8}), back to original order