Unsigned Stirling number of the first kind
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.
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|
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.