Evaluate a Stirling number of the first kind:

Evaluate multiple Stirling numbers:

StirlingS1 threads element–wise over lists:

TraditionalForm formatting:

Plot Stirling numbers of the first kind on a logarithmic scale:

Stirling numbers modulo 2:

Generate the disjoint cycle representations of all permutations of n elements:

Count the number of permutations that have 1, 2, … n disjoint cycles:

The unsigned Stirling number of the first kind counts the number of disjoint cycles:

Plot the average number of cycles in symmetric group elements:

The distribution of the position of the record in the infinite sequence, independent, identically distributed, continuous random variables:

Visualize the probability mass function of the second record:

Code to find the position of the record in a given vector, if any:

Compute positions of the second record in random exponential sequences and compare their histogram to the expected probability mass function:

Generate values from the ordinary generating function:

Generate values from the exponential generating function:

Stirling numbers of the first kind are effectively inverses of Stirling numbers of the second kind:

Calculate large Stirling numbers of the first kind using Cauchy's theorem:

Stirling numbers of the first kind are given by a partial Bell polynomial with sign‐alternating factorial arguments:

StirlingS1 can have large values for moderately sized arguments:

The value at is defined to be 1:

Plot sums of digits:

Determinants of a matrix with Stirling number entries:

Compare with the closed form: