A Cycle Index Spreadsheet

The cycle index of a permutation group is the mean of a list of monomials, one for each permutation in the permutation group. The monomials are themselves each products of an indexed variable—here —where the indices run from 1 to the length of the permutation. Each variable in each monomial is raised to a certain power ranging from 0 to the length of the permutation. That power is the number of cycles in the permutation that have a length equal to the index. By way of example, the permutation {1,3,2,4} has cycles (1)(2,3)(4). There are thus 2 cycles of length 1 and 1 cycle of length 2. The corresponding monomial is thus , which is simplified to . This Demonstration shows how the cycle index is computed in a "spreadsheet" sort of a way. You select the permutation group. You then select the length of the permutation. For each permutation, the Demonstration shows the cycle structure, the number of cycles of each possible length and the corresponding monomial. The bottom-right element of the grid shows the mean of the monomials, which is the cycle index.


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There is a relationship between the cycle index () of a permutation group and cellular automata. For a cellular automaton whose output is invariant under all permutations of the inputs in the permutation group, the number of possible cases the cellular automaton must address is equal to the cycle index of the permutation group after substituting the number of colors in the system for all of the indexed variables (). The number of possible rules is thus equal to . If you hover your mouse over the cycle index, you can see the relationship illustrated.
If you hover your mouse over the terms omitted to conserve space, the Demonstration shows the omitted terms.
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