Identifier
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Mp00207:
Standard tableaux
—horizontal strip sizes⟶
Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤ
Values
[[1]] => [1] => [1,0] => [] => 0
[[1,2]] => [2] => [1,1,0,0] => [] => 0
[[1],[2]] => [1,1] => [1,0,1,0] => [1] => 1
[[1,2,3]] => [3] => [1,1,1,0,0,0] => [] => 0
[[1,3],[2]] => [1,2] => [1,0,1,1,0,0] => [1,1] => 2
[[1,2],[3]] => [2,1] => [1,1,0,0,1,0] => [2] => 1
[[1],[2],[3]] => [1,1,1] => [1,0,1,0,1,0] => [2,1] => 2
[[1,2,3,4]] => [4] => [1,1,1,1,0,0,0,0] => [] => 0
[[1,3,4],[2]] => [1,3] => [1,0,1,1,1,0,0,0] => [1,1,1] => 3
[[1,2,4],[3]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,2] => 2
[[1,2,3],[4]] => [3,1] => [1,1,1,0,0,0,1,0] => [3] => 1
[[1,3],[2,4]] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,1] => 3
[[1,2],[3,4]] => [2,2] => [1,1,0,0,1,1,0,0] => [2,2] => 2
[[1,4],[2],[3]] => [1,1,2] => [1,0,1,0,1,1,0,0] => [2,2,1] => 3
[[1,3],[2],[4]] => [1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,1] => 3
[[1,2],[3],[4]] => [2,1,1] => [1,1,0,0,1,0,1,0] => [3,2] => 2
[[1],[2],[3],[4]] => [1,1,1,1] => [1,0,1,0,1,0,1,0] => [3,2,1] => 3
[[1,2,3,4,5]] => [5] => [1,1,1,1,1,0,0,0,0,0] => [] => 0
[[1,3,4,5],[2]] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => 4
[[1,2,4,5],[3]] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 3
[[1,2,3,5],[4]] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 2
[[1,2,3,4],[5]] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4] => 1
[[1,3,5],[2,4]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 4
[[1,2,5],[3,4]] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,2,2] => 3
[[1,3,4],[2,5]] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 4
[[1,2,4],[3,5]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 3
[[1,2,3],[4,5]] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,3] => 2
[[1,4,5],[2],[3]] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => 4
[[1,3,5],[2],[4]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 4
[[1,2,5],[3],[4]] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 3
[[1,3,4],[2],[5]] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [4,1,1,1] => 4
[[1,2,4],[3],[5]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 3
[[1,2,3],[4],[5]] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [4,3] => 2
[[1,4],[2,5],[3]] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 4
[[1,3],[2,5],[4]] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,3,1,1] => 4
[[1,2],[3,5],[4]] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [3,3,2] => 3
[[1,3],[2,4],[5]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 4
[[1,2],[3,4],[5]] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [4,2,2] => 3
[[1,5],[2],[3],[4]] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => 4
[[1,4],[2],[3],[5]] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,2,2,1] => 4
[[1,3],[2],[4],[5]] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [4,3,1,1] => 4
[[1,2],[3],[4],[5]] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [4,3,2] => 3
[[1],[2],[3],[4],[5]] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => 4
[[1,2,3,4,5,6]] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [] => 0
[[1,3,4,5,6],[2]] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => 5
[[1,2,4,5,6],[3]] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => 4
[[1,2,3,5,6],[4]] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 3
[[1,2,3,4,6],[5]] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => 2
[[1,2,3,4,5],[6]] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5] => 1
[[1,3,5,6],[2,4]] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 5
[[1,2,5,6],[3,4]] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,2,2,2] => 4
[[1,3,4,6],[2,5]] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 5
[[1,2,4,6],[3,5]] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 4
[[1,2,3,6],[4,5]] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 3
[[1,3,4,5],[2,6]] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 5
[[1,2,4,5],[3,6]] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => 4
[[1,2,3,5],[4,6]] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [5,3,3] => 3
[[1,2,3,4],[5,6]] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,4] => 2
[[1,4,5,6],[2],[3]] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => 5
[[1,3,5,6],[2],[4]] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 5
[[1,2,5,6],[3],[4]] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => 4
[[1,3,4,6],[2],[5]] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 5
[[1,2,4,6],[3],[5]] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 4
[[1,2,3,6],[4],[5]] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [4,4,3] => 3
[[1,3,4,5],[2],[6]] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [5,1,1,1,1] => 5
[[1,2,4,5],[3],[6]] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => 4
[[1,2,3,5],[4],[6]] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [5,3,3] => 3
[[1,2,3,4],[5],[6]] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,4] => 2
[[1,3,5],[2,4,6]] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 5
[[1,2,5],[3,4,6]] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => 4
[[1,3,4],[2,5,6]] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 5
[[1,2,4],[3,5,6]] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 4
[[1,2,3],[4,5,6]] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,3,3] => 3
[[1,4,6],[2,5],[3]] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 5
[[1,3,6],[2,5],[4]] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [3,3,3,1,1] => 5
[[1,2,6],[3,5],[4]] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [3,3,3,2] => 4
[[1,3,6],[2,4],[5]] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 5
[[1,2,6],[3,4],[5]] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 4
[[1,4,5],[2,6],[3]] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 5
[[1,3,5],[2,6],[4]] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 5
[[1,2,5],[3,6],[4]] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [5,3,3,2] => 4
[[1,3,4],[2,6],[5]] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [4,4,1,1,1] => 5
[[1,2,4],[3,6],[5]] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [4,4,2,2] => 4
[[1,2,3],[4,6],[5]] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [4,4,3] => 3
[[1,3,5],[2,4],[6]] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 5
[[1,2,5],[3,4],[6]] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [5,2,2,2] => 4
[[1,3,4],[2,5],[6]] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 5
[[1,2,4],[3,5],[6]] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [5,4,2,2] => 4
[[1,2,3],[4,5],[6]] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [5,3,3] => 3
[[1,5,6],[2],[3],[4]] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => 5
[[1,4,6],[2],[3],[5]] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 5
[[1,3,6],[2],[4],[5]] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [4,4,3,1,1] => 5
[[1,2,6],[3],[4],[5]] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [4,4,3,2] => 4
[[1,4,5],[2],[3],[6]] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [5,2,2,2,1] => 5
[[1,3,5],[2],[4],[6]] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 5
[[1,2,5],[3],[4],[6]] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [5,3,3,2] => 4
[[1,3,4],[2],[5],[6]] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [5,4,1,1,1] => 5
[[1,2,4],[3],[5],[6]] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [5,4,2,2] => 4
[[1,2,3],[4],[5],[6]] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [5,4,3] => 3
[[1,4],[2,5],[3,6]] => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,4,2,2,1] => 5
[[1,3],[2,5],[4,6]] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [5,3,3,1,1] => 5
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Description
The length of the partition.
Map
bounce path
Description
The bounce path determined by an integer composition.
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
horizontal strip sizes
Description
The composition of horizontal strip sizes.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.
We associate to a standard Young tableau $T$ the composition $(c_1,\dots,c_k)$, such that $k$ is minimal and the numbers $c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in $T$ for all $i$.
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