PermuNim : an impartial game of permutation avoidance.




Parton, Kristin

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PermuNim is an impartial combinatorial game played on a board of squares where each player takes turns playing in rows and columns of the board which have not been played in, avoiding zero or more permutations. The game comes to an end when neither player can move. The first player unable to move on his or her turn loses the game. Many researchers have investigated combinatorial game theory as well as the idea of permutation pattern avoidance. PermuNim combines both of these ideas. When (12) or (1) is the forbidden permutation in PermuNim, or when the forbidden permutation is 'loses' in size to that of the smallest of the two dimensions of the board, we can say a great deal about the value of the game. For other permutations, the values of the options seem much more chaotic. Even (123) is chaotic as evidenced by our data in the appendix. We investigate the trend for even height boards which are `wide enough' to have options with all odd values and vice versa but we don't believe that this to be true in general. If a PermuNim board is stretched by adding columns, sometimes the value of the position is affected. We find that when any permutation is avoided and t moves have been made, as long at 2m columns are available together, there is a place where any number of columns may be added to the board without affecting the value of the position. We suspect that the number of columns necessary may be much lower for some permutations.



Permutations, Games