# Expected number of cycles of permutation equals harmonic number of degree

## Statement

Suppose  is a natural number. Consider the uniform distribution on the symmetric group of degree . Then, the expected number of cycles in the cycle decomposition of a permutation chosen according to the uniform distribution is equal to the harmonic number  of , where:



Note that , where  is the Euler-Mascheroni constant, and its value is approximately  (or very close to ). Also, . Thus, for  large enough,  can be approximated additively as  and multiplicatively as .

## Particular cases

  (equals expected number of cycles) 
1 1 0
2 1.5 = 3/2 0.6931...
3 1.833.. = 11/6 1.0986...
4 2.0833.. = 25/12 1.3862...
5 2.2833.. = 137/60 1.6094...