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Definition (The Game of Permutation Scoring also called Generate/Don't Generate)
Just as in the Simple Shift Repetition Game, we can play a similar game with any set of permutations of the same set $X$. Let $H$ be a set of permutations of a set $X$. We'll generally assume that the set $X$ is finite so that the game is guaranteed to end. However, you can play this game with an infinite set $X$.
Here are the rules.
- The first player picks an element $\sigma_1\in H$. They then remove from $H$ the span of $\{\sigma_1\}$ (so everything generated by $\sigma_1$).
- The second player now chooses an element $\sigma_2\in H$ that hasn't yet been removed. They then remove from $H$ any element in the span of $\{\sigma_1,\sigma_2\}$. At this stage, any permutation that can be written as a composition combination of the permutations $\sigma_1$ and $\sigma_2$ has been removed.
- Players alternate taking turns, choosing an element $\sigma_k\in H$ that hasn't been removed in a previous stage, and then removing from $H$ any element in the span of $\{\sigma_1,\sigma_2,\ldots,\sigma_k\}$.
- Whoever takes the last element of $H$ wins.
There are several variations:
- The game can also be played as a misere game, where whoever takes the last element of $H$ wins.
- You win by taking the last element. However, you cannot take the last element unless there are no other legal choices. So you can't try to win, rather you must be forced to win.
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