# Unsigned Stirling number of the first kind

## Definition

The unsigned Stirling number of the first kind $|s(n,k)|$, also denoted $S_1(n,k)$ or $c(n,k)$, is defined as the number of elements in the symmetric group of degree $n$ whose cycle decomposition has exactly $k$ 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 $(n,k)$ Value of $|s(n,k)|$ Probability, i.e., value of $|s(n,k)|/n!$ Explanation
$(n,1)$ $(n - 1)!$ $1/n$ must be a cyclic permutation with one cycle of size $n$, then use conjugacy class size formula for symmetric group.
$(n,n)$ $1$ $1/n!$ can only be the identity permutation.
$(n,n-1)$ $\binom{n}{2} = \frac{n(n-1)}{2}$ $\frac{1}{2[(n-2)!]}$ must be a transposition, determined by picking a subset of size two that gets transposed.
$(n,2)$, $n$ odd $(n - 1)! H_{n-1}$, where $H_{n-1}$ is the harmonic number, i.e., the sum of reciprocals of first $n - 1$ natural numbers $H_{n-1}/n \approx (\log n)/n$

## Particular cases

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

$n$ $n!$ $|s(n,1)|$ $|s(n,2)|$ $|s(n,3)|$ $|s(n,4)|$ $|s(n,5)|$ $|s(n,6)|$ $|s(n,7)|$ $|s(n,8)|$
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