gives the Stirling number of the first kind .


  • Integer mathematical function, suitable for both symbolic and numerical manipulation.
  • gives the number of permutations of elements that contain exactly cycles.
  • StirlingS1 automatically threads over lists.


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Basic Examples  (1)

Scope  (2)

StirlingS1 threads elementwise over lists:

TraditionalForm formatting:

Applications  (3)

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

Plot the average number of cycles in symmetric group elements:

The distribution of the position of the ^(th) 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 ^(th) record in a given vector, if any:

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

Properties & Relations  (4)

Generate values from the 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 signalternating factorial arguments:

Possible Issues  (2)

StirlingS1 can have large values for moderately sized arguments:

The value at is defined to be 1:

Neat Examples  (2)

Plot sums of digits:

Introduced in 1988