Supposing I have a fair die and record the rolls, but ignore and reroll any immediate repeats. For example, I roll:
3 2 1 (1) 5 6 (6) (6) 3 1
And I record:
3 2 1 5 6 3 1
...and so forth, ignoring (1) and (6) (6) and in the process of generating a non-repeating sequence.
Are there special terms of art in mathematics for this kind of random non-repeating sequences, and are their concepts or equations for their properties?
For example, it seems like the number of sequences would be 6 * (6-1) * (6-1) ... and that if next roll was not allowed to match any number of previous rolls x, then the number of possible sequences would be n * (n-1) * (n-2) ... * (n-x) * (n-x) * (n-x) ...
As an illustrative example, I'm thinking of the case of a music player with a large number of songs. Rather than shuffling, it randomly chooses a song, then rejects it only if it is an immediate repeat. This concept could then be extended to rejecting a song that was one of the x previous songs. This isn't a shuffle, and it isn't purely random. Are there concise ways of describing it?
It's a Markov chain. In particular, it's a random walk on the graph $K_6$.