Identifier
Values
[1,0,1,0] => [1,0,1,0] => [1,1,0,0] => [2,1] => 0
[1,1,0,0] => [1,1,0,0] => [1,0,1,0] => [1,2] => 0
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [3,2,1] => 0
[1,0,1,1,0,0] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [2,3,1] => 0
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [1,0,1,0,1,0] => [1,2,3] => 0
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [2,3,1] => 0
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [1,1,0,0,1,0] => [2,1,3] => 0
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [3,4,2,1] => 0
[1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 0
[1,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [3,4,2,1] => 0
[1,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => 0
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 1
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 0
[1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => 0
[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 0
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0
[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => 0
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 0
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 0
[1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 0
[1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 0
[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 0
[1,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 1
[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 0
[1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 0
[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 0
[1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0
[1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 0
[1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 1
[1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 1
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0
[1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 0
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0
[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 0
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 1
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 0
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 0
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 0
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 0
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 0
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 0
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 0
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 0
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 0
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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
Knuth-Krattenthaler
Description
The map that sends the Dyck path to a 321-avoiding permutation, then applies the Robinson-Schensted correspondence and finally interprets the first row of the insertion tableau and the second row of the recording tableau as up steps.
Interpreting a pair of two-row standard tableaux of the same shape as a Dyck path is explained by Knuth in [1, pp. 60].
Krattenthaler's bijection between Dyck paths and $321$-avoiding permutations used is Mp00119to 321-avoiding permutation (Krattenthaler), see [2].
This is the inverse of the map Mp00127left-to-right-maxima to Dyck path that interprets the left-to-right maxima of the permutation obtained from Mp00024to 321-avoiding permutation as a Dyck path.
Map
to 312-avoiding permutation
Description
Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).
Map
bounce path
Description
Sends a Dyck path $D$ of length $2n$ to its bounce path.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and travelling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again to hit $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.