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

From Groupprops

## 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 .

See also probability distribution of number of cycles of permutation.

## 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... |