Function Repository Resource:

InvolutionCount

Count the number of involutions

Contributed by: Wolfram Staff (original content by Sriram V. Pemmaraju and Steven S. Skiena)

ResourceFunction["InvolutionCount"][n]

gives the number of involutions on n elements.

Details and Options

An involution on n elements is a permutation of length n that is its own inverse, i.e. a permutation σ with Length[σ]==n such that σ² is the identity permutation.

Involution numbers are also known as telephone numbers, as they count the ways n telephone lines can be connected to each other with each line connecting to at most one other line.