Identifier
Values
[2] => [1,0,1,0] => [1,2] => [1,2] => 0
[1,1] => [1,1,0,0] => [2,1] => [2,1] => 0
[3] => [1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 0
[2,1] => [1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 1
[1,1,1] => [1,1,0,1,0,0] => [2,3,1] => [3,1,2] => 0
[4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 2
[2,2] => [1,1,1,0,0,0] => [3,2,1] => [3,2,1] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,1,2,3] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 3
[3,2] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,4,3,2] => 2
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,3,4] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => [4,1,3,2] => 1
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,5,4,3] => 4
[3,3] => [1,1,1,0,1,0,0,0] => [3,4,2,1] => [4,2,1,3] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,5,2,4,3] => 3
[2,2,2] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,2,1] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [5,1,3,2,4] => 1
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [1,5,3,2,4] => 2
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [5,1,4,2,3] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [5,1,4,3,2] => 2
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [5,2,1,3,4] => 0
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [5,4,2,1,3] => 0
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [5,3,2,1,4] => 0
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,4,3,2,1] => 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
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding 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
invert Laguerre heap
Description
The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.