You can represent permutations using disjoint cycles by creating a function that constructs these cycles and applies them to a set. Disjoint cycles are a way to express permutations where each element moves to another element in a cycle, and elements not mentioned in any cycle are fixed.
You can follow the given below steps to create and use disjoint cycles in MATLAB:
- Specify the cycles you want to use. Each cycle is a list of elements that are permuted among themselves.
- Write a function to apply these cycles to a set of numbers, typically from 1 to n.
- Elements not included in any cycle remain unchanged.
Below is the sample code that might help you in getting started:
function perm = applyDisjointCycles(n, cycles)
% n: Total number of elements in the set (e.g., 16)
% cycles: Cell array where each cell contains a vector representing a cycle
% Initialize the permutation as an identity permutation
perm = 1:n;
% Apply each cycle to the permutation
for i = 1:length(cycles)
cycle = cycles{i};
numElements = length(cycle);
for j = 1:numElements
% Map each element to the next in the cycle, wrapping around at the end
nextIndex = mod(j, numElements) + 1;
perm(cycle(j)) = cycle(nextIndex);
end
end
end
n = 16;
cyclesA = {[1, 3, 5, 7, 9, 11]};
cyclesB = {[2, 4, 6, 8, 10, 12, 14, 16]};
permA = applyDisjointCycles(n, cyclesA);
permB = applyDisjointCycles(n, cyclesB);
disp('Permutation A:');
disp(permA);
disp('Permutation B:');
disp(permB);
Hope this helps!
