Ewens Permutation Cycle Law
Weight permutation cycles and connect enumeration to a rising-factorial probability law.
Free theory articleWeight permutation cycles and connect enumeration to a rising-factorial probability law.
Concept & interpretation
The Ewens sampling formula assigns probabilities to partitions of a finite set, particularly arising in population genetics and the study of random permutations. When viewed as a distribution over the symmetric group of permutations of length n, it assigns a weight proportional to θ raised to the number of cycles in the permutation's disjoint cycle decomposition.
experiment
Open the module and observe how the probability distribution over cycle counts shifts as you change the parameter θ.
- Set Permutation size to 4 and Theta to 1. Notice the expected cycles is about 2.083.
- Increase Theta to 4. Observe that the distribution shifts towards a higher number of cycles.
- Set Permutation size to 0. Verify that the boundary case yields 0 expected cycles and variance.