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2020-09-12, 03:25   #23
0scar

Jan 2020

C16 Posts

Quote:
 Originally Posted by SmartMersenne Yes, the identity mapping is counted as a permutation. And the mapping can have two disconnected cycles, but I doubt that it would yield a solution. Here is how you should define the mapping: [1,2,0,4,3] where the indices are the original numbers and the entries are the result of the permutation.
I didn't know the chosen "cycle notation" too, I always used the "one-line notation" described by SmartMersenne.
Wikipedia has an useful paragraph about them (https://en.wikipedia.org/wiki/Permutation#Notations).
If we want to arrange the numbers from 0 to 6 in descending order, we have:
the one-line notation [6,5,4,3,2,1,0],
the cycle notation (0,6)(1,5)(2,4)(3),
but 1-cycles like (3) are often omitted.

For all RPS(n) games with n<=5, non-trivial automorphisms have exactly one cycle (more precisely, one n-cycle), but this property doesn't hold in general.
I guess it isn't spoiling to show the following RPS(7) game, far from being optimal:

0 -> 1, 2, 3
1 -> 4, 5, 6
2 -> 1, 4, 5
3 -> 1, 2, 6
4 -> 0, 3, 6
5 -> 0, 3, 4
6 -> 0, 2, 5

[2,4,5,1,3,0,6], or (0,2,5)(1,4,3)(6)
[5,3,0,4,1,2,6], or (0,5,2)(1,3,4)(6)

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