# Unsigned Stirling number of the first kind

From Groupprops

## Definition

The **unsigned Stirling number of the first kind** , also denoted or , is defined as the number of elements in the symmetric group of degree whose cycle decomposition has exactly cycles. (Here, each fixed point is treated as a cycle of size one).

The unsigned Stirling numbers play a role in the probability distribution of number of cycles of permutations.

## General information

### Formulas in special cases

Case for | Value of | Probability, i.e., value of | Explanation |
---|---|---|---|

must be a cyclic permutation with one cycle of size , then use conjugacy class size formula for symmetric group. | |||

can only be the identity permutation. | |||

must be a transposition, determined by picking a subset of size two that gets transposed. | |||

, odd | , where is the harmonic number, i.e., the sum of reciprocals of first natural numbers |

## Particular cases

Note that the unsigned Stirling numbers of the first kind for fixed form a unimodal (single-peaked) sequence in , i.e., they are first increasing and then decreasing.

1 | 1 | 1 | |||||||

2 | 2 | 1 | 1 | ||||||

3 | 6 | 2 | 3 | 1 | |||||

4 | 24 | 6 | 11 | 6 | 1 | ||||

5 | 120 | 24 | 50 | 35 | 10 | 1 | |||

6 | 720 | 120 | 274 | 225 | 85 | 15 | 1 | ||

7 | 5040 | 720 | 1764 | 1624 | 735 | 175 | 21 | 1 | |

8 | 40320 | 5040 | 13068 | 13132 | 6769 | 1960 | 322 | 28 | 1 |