How this calculation works
Count permutations with exactly k consecutive increasing pairs.
THE FORMULA
A(n,k)=(k+1)A(n−1,k)+(n−k)A(n−1,k−1)A worked example
Eleven permutations of 1 through 4 have exactly one ascent.
Good to know
An ascent uses adjacent positions.
Reference: NIST DLMF, Functions of Number Theory (opens in a new tab) ↗
Read the result
Count permutations with exactly k consecutive increasing pairs.
The primary result is labeled “Permutations”. Keep the input definitions in view when comparing this result with another method.
Try a related calculation
An ascent uses adjacent positions.
For a related exercise, try the greatest common divisor.
A few questions, answered
How can I check the default inputs by hand?
Eleven permutations of 1 through 4 have exactly one ascent.
Which rule does this calculator apply?
A(n,k)=(k+1)A(n−1,k)+(n−k)A(n−1,k−1)