Identifier
Values
[1,2] => [1,2] => [1,2] => [2,1] => 0
[2,1] => [1,2] => [1,2] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [3,1,2] => 0
[2,1,3] => [1,3,2] => [2,3,1] => [3,1,2] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[3,1,2] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[3,2,1] => [1,2,3] => [1,2,3] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,3,4,1] => [3,4,1,2] => 0
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => [3,4,2,1] => 0
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => [3,2,1,4] => 0
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => [3,2,1,4] => 0
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => [3,2,1,4] => 0
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => [3,2,1,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[2,4,1,3] => [1,3,2,4] => [2,3,1,4] => [3,4,2,1] => 0
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => [3,4,1,2] => 0
[3,1,2,4] => [1,2,4,3] => [2,3,4,1] => [3,4,1,2] => 0
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => [3,2,1,4] => 0
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => [3,2,1,4] => 0
[3,2,4,1] => [1,2,4,3] => [2,3,4,1] => [3,4,1,2] => 0
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[4,1,3,2] => [1,3,2,4] => [2,3,1,4] => [3,4,2,1] => 0
[4,2,1,3] => [1,3,2,4] => [2,3,1,4] => [3,4,2,1] => 0
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => [3,4,5,2,1] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => [3,4,1,5,2] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => [3,4,2,5,1] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => [4,5,3,1,2] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => [3,5,2,4,1] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,1,3,4] => [3,1,4,5,2] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,4,5,3] => [3,2,5,1,4] => 2
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,4,5,3] => [3,2,5,1,4] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,3,2] => 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,3,2] => 1
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,5,3,4] => [3,2,1,5,4] => 3
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,5,3,4] => [3,2,1,5,4] => 3
[1,5,2,3,4] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[1,5,2,4,3] => [1,5,2,4,3] => [3,2,5,4,1] => [4,3,1,2,5] => 0
[1,5,3,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => [4,3,1,2,5] => 0
[1,5,3,4,2] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[1,5,4,2,3] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[1,5,4,3,2] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[2,1,3,4,5] => [1,3,4,5,2] => [2,5,1,3,4] => [3,1,4,5,2] => 1
[2,1,3,5,4] => [1,3,5,2,4] => [2,1,4,5,3] => [3,2,5,1,4] => 2
[2,1,4,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[2,1,4,5,3] => [1,4,5,2,3] => [2,1,5,3,4] => [3,2,1,5,4] => 3
[2,1,5,3,4] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[2,1,5,4,3] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[2,3,1,4,5] => [1,4,5,2,3] => [2,1,5,3,4] => [3,2,1,5,4] => 3
[2,3,1,5,4] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[2,3,4,1,5] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[2,3,5,1,4] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[2,4,1,3,5] => [1,3,5,2,4] => [2,1,4,5,3] => [3,2,5,1,4] => 2
[2,4,1,5,3] => [1,5,2,4,3] => [3,2,5,4,1] => [4,3,1,2,5] => 0
[2,4,3,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => [4,3,1,2,5] => 0
[2,4,3,5,1] => [1,2,4,3,5] => [2,3,4,1,5] => [3,4,5,2,1] => 0
[2,4,5,1,3] => [1,3,2,4,5] => [2,3,1,4,5] => [3,4,2,5,1] => 0
[2,4,5,3,1] => [1,2,4,5,3] => [2,3,5,1,4] => [3,4,1,5,2] => 1
[2,5,1,3,4] => [1,3,4,2,5] => [2,4,1,3,5] => [3,5,2,4,1] => 1
[2,5,1,4,3] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,3,2] => 1
[2,5,3,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,3,2] => 1
[2,5,3,4,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[2,5,4,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => [4,5,3,1,2] => 0
[2,5,4,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[3,1,2,4,5] => [1,2,4,5,3] => [2,3,5,1,4] => [3,4,1,5,2] => 1
[3,1,2,5,4] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[3,1,4,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,3,2] => 1
[3,1,4,5,2] => [1,4,5,2,3] => [2,1,5,3,4] => [3,2,1,5,4] => 3
[3,1,5,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => [4,3,1,2,5] => 0
[3,1,5,4,2] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[3,2,1,4,5] => [1,4,5,2,3] => [2,1,5,3,4] => [3,2,1,5,4] => 3
[3,2,1,5,4] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[3,2,4,1,5] => [1,5,2,4,3] => [3,2,5,4,1] => [4,3,1,2,5] => 0
[3,2,4,5,1] => [1,2,4,5,3] => [2,3,5,1,4] => [3,4,1,5,2] => 1
[3,2,5,1,4] => [1,4,2,5,3] => [3,4,5,1,2] => [4,5,1,3,2] => 1
[3,2,5,4,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[3,4,1,2,5] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[3,4,1,5,2] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[3,4,2,1,5] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[3,4,2,5,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[3,5,1,2,4] => [1,2,4,3,5] => [2,3,4,1,5] => [3,4,5,2,1] => 0
[3,5,1,4,2] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[3,5,2,1,4] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
>>> Load all 152 entries. <<<
[3,5,2,4,1] => [1,2,4,3,5] => [2,3,4,1,5] => [3,4,5,2,1] => 0
[3,5,4,1,2] => [1,2,3,5,4] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[3,5,4,2,1] => [1,2,3,5,4] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[4,1,2,3,5] => [1,2,3,5,4] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[4,1,2,5,3] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[4,1,3,2,5] => [1,3,2,5,4] => [3,4,2,5,1] => [4,5,3,1,2] => 0
[4,1,3,5,2] => [1,3,5,2,4] => [2,1,4,5,3] => [3,2,5,1,4] => 2
[4,1,5,2,3] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[4,1,5,3,2] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[4,2,1,3,5] => [1,3,5,2,4] => [2,1,4,5,3] => [3,2,5,1,4] => 2
[4,2,1,5,3] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[4,2,3,1,5] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[4,2,3,5,1] => [1,2,3,5,4] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[4,2,5,1,3] => [1,3,2,5,4] => [3,4,2,5,1] => [4,5,3,1,2] => 0
[4,2,5,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[4,3,1,2,5] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[4,3,1,5,2] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[4,3,2,1,5] => [1,5,2,3,4] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[4,3,2,5,1] => [1,2,5,3,4] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[4,3,5,1,2] => [1,2,3,5,4] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[4,3,5,2,1] => [1,2,3,5,4] => [2,3,4,5,1] => [3,4,5,1,2] => 0
[4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[4,5,1,3,2] => [1,3,2,4,5] => [2,3,1,4,5] => [3,4,2,5,1] => 0
[4,5,2,1,3] => [1,3,2,4,5] => [2,3,1,4,5] => [3,4,2,5,1] => 0
[4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[5,1,2,4,3] => [1,2,4,3,5] => [2,3,4,1,5] => [3,4,5,2,1] => 0
[5,1,3,2,4] => [1,3,2,4,5] => [2,3,1,4,5] => [3,4,2,5,1] => 0
[5,1,3,4,2] => [1,3,4,2,5] => [2,4,1,3,5] => [3,5,2,4,1] => 1
[5,1,4,2,3] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[5,1,4,3,2] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[5,2,1,3,4] => [1,3,4,2,5] => [2,4,1,3,5] => [3,5,2,4,1] => 1
[5,2,1,4,3] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[5,2,3,1,4] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[5,2,4,1,3] => [1,3,2,4,5] => [2,3,1,4,5] => [3,4,2,5,1] => 0
[5,2,4,3,1] => [1,2,4,3,5] => [2,3,4,1,5] => [3,4,5,2,1] => 0
[5,3,1,2,4] => [1,2,4,3,5] => [2,3,4,1,5] => [3,4,5,2,1] => 0
[5,3,1,4,2] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[5,3,2,1,4] => [1,4,2,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2
[5,3,2,4,1] => [1,2,4,3,5] => [2,3,4,1,5] => [3,4,5,2,1] => 0
[5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[5,4,1,3,2] => [1,3,2,4,5] => [2,3,1,4,5] => [3,4,2,5,1] => 0
[5,4,2,1,3] => [1,3,2,4,5] => [2,3,1,4,5] => [3,4,2,5,1] => 0
[5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The Castelnuovo-Mumford regularity of a permutation.
The Castelnuovo-Mumford regularity of a permutation $\sigma$ is the Castelnuovo-Mumford regularity of the matrix Schubert variety $X_\sigma$.
Equivalently, it is the difference between the degrees of the Grothendieck polynomial and the Schubert polynomial for $\sigma$. It can be computed by subtracting the Coxeter length St000018The number of inversions of a permutation. from the Rajchgot index St001759The Rajchgot index of a permutation..
Map
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
Map
Lehmer code rotation
Description
Sends a permutation $\pi$ to the unique permutation $\tau$ (of the same length) such that every entry in the Lehmer code of $\tau$ is cyclically one larger than the Lehmer code of $\pi$.