Probability distribution of number of cycles of permutations

Definition
Let $$n$$ be a natural number. Denote by $$S_n$$ the symmetric group of degree $$n$$. Consider the uniform probability measure on $$S_n$$ (i.e., the normalized counting measure, under which the measure of any subset is the quotient of the size of the subset by $$n!$$). The probability distribution of number of cycles of permutations is the probability distribution that assigns, to all numbers $$1 \le k \le n$$, the probability that a permutation picked uniformly at random from $$S_n$$ has exactly $$k$$ cycles in its cycle decomposition.

Note that in the count of cycles, each fixed point is treated as one cycle. Thus, the identity permutation has $$n$$ cycles whereas a cyclic permutation that moves all elements involves $$1$$ cycle.

The probability distribution assigns the following value to each $$k$$

$$\! \frac{|s(n,k)|}{n!}$$

where $$|s(n,k)|$$ is the unsigned Stirling number of the first kind, and occurs as an important combinatorial quantity. It can also be defined as the sum, over all unordered partitions of $$n$$ into $$k$$ parts, of the sizes of the conjugacy classes corresponding to each part.

Particular cases
The unsigned Stirling number of the first kind for any given $$n$$ form a unimodal sequence, hence the corresponding probability distribution is also unimodal (i.e., single-peaked).

Here is the table of unsigned Stirling numbers:

Here is the table of probability values:

Computer package implementation
All unbounded variables are either defined beforehand or are replaced by actual numerical values.