Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001741: Permutations ⟶ ℤ
Values
[1,0,1,0] => [1] => [[1]] => [1] => 1
[1,0,1,0,1,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 2
[1,0,1,1,0,0] => [1,1] => [[1],[2]] => [2,1] => 1
[1,1,0,0,1,0] => [2] => [[1,2]] => [1,2] => 1
[1,1,0,1,0,0] => [1] => [[1]] => [1] => 1
[1,0,1,0,1,0,1,0] => [3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => 3
[1,0,1,0,1,1,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 2
[1,0,1,1,0,0,1,0] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 2
[1,0,1,1,0,1,0,0] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 2
[1,0,1,1,1,0,0,0] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[1,1,0,0,1,0,1,0] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2
[1,1,0,0,1,1,0,0] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 2
[1,1,0,1,0,1,0,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 2
[1,1,0,1,1,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 1
[1,1,1,0,0,0,1,0] => [3] => [[1,2,3]] => [1,2,3] => 1
[1,1,1,0,0,1,0,0] => [2] => [[1,2]] => [1,2] => 1
[1,1,1,0,1,0,0,0] => [1] => [[1]] => [1] => 1
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => 2
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => 2
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 1
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 2
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => 3
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => 2
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => 2
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 2
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 2
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 2
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => [[1,2,3,4]] => [1,2,3,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [3] => [[1,2,3]] => [1,2,3] => 1
[1,1,1,1,0,0,1,0,0,0] => [2] => [[1,2]] => [1,2] => 1
[1,1,1,1,0,1,0,0,0,0] => [1] => [[1]] => [1] => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => 2
[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] => 1
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => 2
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => 2
[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] => 2
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 2
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => 3
[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] => 2
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => 2
[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] => 2
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => 2
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 2
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => 2
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => 2
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => 2
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 2
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [5] => [[1,2,3,4,5]] => [1,2,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[1,2,3,4]] => [1,2,3,4] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[1,2,3]] => [1,2,3] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [[1,2]] => [1,2] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [[1]] => [1] => 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => 2
[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] => 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => 2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => 2
[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] => 2
[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] => 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => 3
[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] => 2
[1,1,1,1,0,1,1,0,0,0,1,0,0,0] => [4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => 2
[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] => 2
[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] => 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [3,2,1] => 1
[1,1,1,1,1,0,0,0,1,1,0,0,0,0] => [3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,0] => [4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => 2
[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] => 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [3,4,1,2] => 2
[1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => 2
[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] => 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [[1,3,4],[2]] => [2,1,3,4] => 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [[1,3],[2]] => [2,1,3] => 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [[1],[2]] => [2,1] => 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => 1
[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] => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => [[1,2,3,4]] => [1,2,3,4] => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [[1,2,3]] => [1,2,3] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [[1,2]] => [1,2] => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => [[1]] => [1] => 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => 1
[1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => 2
[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] => 1
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => 2
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Description
The largest integer such that all patterns of this size are contained in the permutation.
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
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
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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