Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤ
Values
[1,0,1,0] => [1] => [[1]] => 0
[1,0,1,0,1,0] => [2,1] => [[1,3],[2]] => 0
[1,0,1,1,0,0] => [1,1] => [[1],[2]] => 1
[1,1,0,0,1,0] => [2] => [[1,2]] => 0
[1,1,0,1,0,0] => [1] => [[1]] => 0
[1,0,1,0,1,0,1,0] => [3,2,1] => [[1,3,6],[2,5],[4]] => 0
[1,0,1,0,1,1,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => 1
[1,0,1,1,0,0,1,0] => [3,1,1] => [[1,4,5],[2],[3]] => 0
[1,0,1,1,0,1,0,0] => [2,1,1] => [[1,4],[2],[3]] => 0
[1,0,1,1,1,0,0,0] => [1,1,1] => [[1],[2],[3]] => 2
[1,1,0,0,1,0,1,0] => [3,2] => [[1,2,5],[3,4]] => 0
[1,1,0,0,1,1,0,0] => [2,2] => [[1,2],[3,4]] => 1
[1,1,0,1,0,0,1,0] => [3,1] => [[1,3,4],[2]] => 0
[1,1,0,1,0,1,0,0] => [2,1] => [[1,3],[2]] => 0
[1,1,0,1,1,0,0,0] => [1,1] => [[1],[2]] => 1
[1,1,1,0,0,0,1,0] => [3] => [[1,2,3]] => 0
[1,1,1,0,0,1,0,0] => [2] => [[1,2]] => 0
[1,1,1,0,1,0,0,0] => [1] => [[1]] => 0
[1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => 0
[1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => 2
[1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => 1
[1,0,1,1,0,1,0,0,1,0] => [4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => 0
[1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 0
[1,0,1,1,0,1,1,0,0,0] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 1
[1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => 0
[1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 0
[1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [[1,5],[2],[3],[4]] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => 3
[1,1,0,0,1,0,1,1,0,0] => [3,3,2] => [[1,2,5],[3,4,8],[6,7]] => 1
[1,1,0,0,1,1,0,0,1,0] => [4,2,2] => [[1,2,7,8],[3,4],[5,6]] => 0
[1,1,0,0,1,1,0,1,0,0] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => 0
[1,1,0,0,1,1,1,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => 2
[1,1,0,1,0,0,1,0,1,0] => [4,3,1] => [[1,3,4,8],[2,6,7],[5]] => 0
[1,1,0,1,0,0,1,1,0,0] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => 1
[1,1,0,1,0,1,0,0,1,0] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 0
[1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [[1,3,6],[2,5],[4]] => 0
[1,1,0,1,0,1,1,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => 1
[1,1,0,1,1,0,0,0,1,0] => [4,1,1] => [[1,4,5,6],[2],[3]] => 0
[1,1,0,1,1,0,0,1,0,0] => [3,1,1] => [[1,4,5],[2],[3]] => 0
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => 2
[1,1,1,0,0,0,1,0,1,0] => [4,3] => [[1,2,3,7],[4,5,6]] => 0
[1,1,1,0,0,0,1,1,0,0] => [3,3] => [[1,2,3],[4,5,6]] => 1
[1,1,1,0,0,1,0,0,1,0] => [4,2] => [[1,2,5,6],[3,4]] => 0
[1,1,1,0,0,1,0,1,0,0] => [3,2] => [[1,2,5],[3,4]] => 0
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [[1,2],[3,4]] => 1
[1,1,1,0,1,0,0,0,1,0] => [4,1] => [[1,3,4,5],[2]] => 0
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [[1,3,4],[2]] => 0
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [[1,3],[2]] => 0
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [[1],[2]] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => [[1,2,3,4]] => 0
[1,1,1,1,0,0,0,1,0,0] => [3] => [[1,2,3]] => 0
[1,1,1,1,0,0,1,0,0,0] => [2] => [[1,2]] => 0
[1,1,1,1,0,1,0,0,0,0] => [1] => [[1]] => 0
[1,0,1,1,0,1,1,1,0,0,0,0] => [2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => 0
[1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => 0
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => 0
[1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [[1],[2],[3],[4],[5]] => 4
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => 3
[1,1,0,1,0,1,1,0,1,0,0,0] => [3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => 0
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => 2
[1,1,0,1,1,0,0,1,1,0,0,0] => [3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => 1
[1,1,0,1,1,0,1,0,0,1,0,0] => [4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => 0
[1,1,0,1,1,0,1,0,1,0,0,0] => [3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => 0
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,1,1] => [[1,4],[2,6],[3],[5]] => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => 0
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => 0
[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,1,1] => [[1,5,6],[2],[3],[4]] => 0
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,1,1,1] => [[1,5],[2],[3],[4]] => 0
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => 3
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,3,2] => [[1,2,5],[3,4,8],[6,7]] => 1
[1,1,1,0,0,1,1,0,0,1,0,0] => [4,2,2] => [[1,2,7,8],[3,4],[5,6]] => 0
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,2] => [[1,2,7],[3,4],[5,6]] => 0
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,2,2] => [[1,2],[3,4],[5,6]] => 2
[1,1,1,0,1,0,0,1,0,1,0,0] => [4,3,1] => [[1,3,4,8],[2,6,7],[5]] => 0
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,3,1] => [[1,3,4],[2,6,7],[5]] => 1
[1,1,1,0,1,0,1,0,0,0,1,0] => [5,2,1] => [[1,3,6,7,8],[2,5],[4]] => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [[1,3,6,7],[2,5],[4]] => 0
[1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [[1,3,6],[2,5],[4]] => 0
[1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [[1,3],[2,5],[4]] => 1
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1,1] => [[1,4,5,6,7],[2],[3]] => 0
[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [[1,4,5,6],[2],[3]] => 0
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,1,1] => [[1,4,5],[2],[3]] => 0
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => 2
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => [[1,2,3,4],[5,6,7,8]] => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [5,3] => [[1,2,3,7,8],[4,5,6]] => 0
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [[1,2,3,7],[4,5,6]] => 0
[1,1,1,1,0,0,0,1,1,0,0,0] => [3,3] => [[1,2,3],[4,5,6]] => 1
[1,1,1,1,0,0,1,0,0,0,1,0] => [5,2] => [[1,2,5,6,7],[3,4]] => 0
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,2] => [[1,2,5,6],[3,4]] => 0
[1,1,1,1,0,0,1,0,1,0,0,0] => [3,2] => [[1,2,5],[3,4]] => 0
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[1,2],[3,4]] => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [[1,3,4,5,6],[2]] => 0
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [[1,3,4,5],[2]] => 0
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[1,3,4],[2]] => 0
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[1,3],[2]] => 0
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [[1],[2]] => 1
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Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
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 $(n-1,\ldots,2,1)$.
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.
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