Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001811: Permutations ⟶ ℤ
Values
[1,0,1,0,1,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[1,0,1,1,0,0] => [1,1] => [[1],[2]] => [2,1] => 0
[1,1,0,0,1,0] => [2] => [[1,2]] => [1,2] => 0
[1,0,1,0,1,1,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[1,0,1,1,0,0,1,0] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,0,1,1,0,1,0,0] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[1,0,1,1,1,0,0,0] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[1,1,0,0,1,0,1,0] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 0
[1,1,0,0,1,1,0,0] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 0
[1,1,0,1,0,0,1,0] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[1,1,0,1,0,1,0,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[1,1,0,1,1,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 0
[1,1,1,0,0,0,1,0] => [3] => [[1,2,3]] => [1,2,3] => 0
[1,1,1,0,0,1,0,0] => [2] => [[1,2]] => [1,2] => 0
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 0
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 0
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 0
[1,1,1,1,0,0,0,0,1,0] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,1,0,0] => [3] => [[1,2,3]] => [1,2,3] => 0
[1,1,1,1,0,0,1,0,0,0] => [2] => [[1,2]] => [1,2] => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 0
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 0
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[1,2,3]] => [1,2,3] => 0
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [[1,2]] => [1,2] => 0
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 0
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 0
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 0
[1,1,1,1,1,0,1,0,0,0,1,0,0,0] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 0
[1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [[1,2,3]] => [1,2,3] => 0
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [[1,2]] => [1,2] => 0
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 0
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 0
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 0
[1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 0
[1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [[1,2,3]] => [1,2,3] => 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [[1,2]] => [1,2] => 0
[1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => 0
[1,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 0
[1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 0
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 0
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 0
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 0
[1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 0
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 0
[1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 0
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 0
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 0
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 0
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [4] => [[1,2,3,4]] => [1,2,3,4] => 0
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => [[1,2,3]] => [1,2,3] => 0
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => [[1,2]] => [1,2] => 0
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 0
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => [[1,2]] => [1,2] => 0
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 0
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 1
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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..
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
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
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