Identifier
-
Mp00027:
Dyck paths
—to partition⟶
Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St001686: Gelfand-Tsetlin patterns ⟶ ℤ
Values
[1,0,1,0,1,0] => [2,1] => [[1,3],[2]] => [[2,1,0],[1,1],[1]] => 2
[1,0,1,1,0,0] => [1,1] => [[1],[2]] => [[1,1],[1]] => 1
[1,1,0,0,1,0] => [2] => [[1,2]] => [[2,0],[1]] => 1
[1,0,1,1,0,1,0,0] => [2,1,1] => [[1,4],[2],[3]] => [[2,1,1,0],[1,1,1],[1,1],[1]] => 3
[1,0,1,1,1,0,0,0] => [1,1,1] => [[1],[2],[3]] => [[1,1,1],[1,1],[1]] => 1
[1,1,0,0,1,1,0,0] => [2,2] => [[1,2],[3,4]] => [[2,2,0,0],[2,1,0],[2,0],[1]] => 2
[1,1,0,1,0,0,1,0] => [3,1] => [[1,3,4],[2]] => [[3,1,0,0],[2,1,0],[1,1],[1]] => 3
[1,1,0,1,0,1,0,0] => [2,1] => [[1,3],[2]] => [[2,1,0],[1,1],[1]] => 2
[1,1,0,1,1,0,0,0] => [1,1] => [[1],[2]] => [[1,1],[1]] => 1
[1,1,1,0,0,0,1,0] => [3] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => 1
[1,1,1,0,0,1,0,0] => [2] => [[1,2]] => [[2,0],[1]] => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [[1,1,1,1],[1,1,1],[1,1],[1]] => 1
[1,1,0,1,1,0,1,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [[2,1,1,0],[1,1,1],[1,1],[1]] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [[1,1,1],[1,1],[1]] => 1
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [[1,2],[3,4]] => [[2,2,0,0],[2,1,0],[2,0],[1]] => 2
[1,1,1,0,1,0,0,1,0,0] => [3,1] => [[1,3,4],[2]] => [[3,1,0,0],[2,1,0],[1,1],[1]] => 3
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [[1,3],[2]] => [[2,1,0],[1,1],[1]] => 2
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [[1],[2]] => [[1,1],[1]] => 1
[1,1,1,1,0,0,0,0,1,0] => [4] => [[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => 1
[1,1,1,1,0,0,0,1,0,0] => [3] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => 1
[1,1,1,1,0,0,1,0,0,0] => [2] => [[1,2]] => [[2,0],[1]] => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [[1,1,1,1],[1,1,1],[1,1],[1]] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [[2,1,1,0],[1,1,1],[1,1],[1]] => 3
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [[1,1,1],[1,1],[1]] => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [[2,2,0,0],[2,1,0],[2,0],[1]] => 2
[1,1,1,1,0,1,0,0,1,0,0,0] => [3,1] => [[1,3,4],[2]] => [[3,1,0,0],[2,1,0],[1,1],[1]] => 3
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [[1,3],[2]] => [[2,1,0],[1,1],[1]] => 2
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [[1],[2]] => [[1,1],[1]] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [[1,2]] => [[2,0],[1]] => 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [[1,1,1,1],[1,1,1],[1,1],[1]] => 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [[2,1,1,0],[1,1,1],[1,1],[1]] => 3
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [[1,1,1],[1,1],[1]] => 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [[2,2,0,0],[2,1,0],[2,0],[1]] => 2
[1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [[1,3,4],[2]] => [[3,1,0,0],[2,1,0],[1,1],[1]] => 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [[1,3],[2]] => [[2,1,0],[1,1],[1]] => 2
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [[1],[2]] => [[1,1],[1]] => 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,0] => [4] => [[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [[1,2]] => [[2,0],[1]] => 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [[1,1,1,1],[1,1,1],[1,1],[1]] => 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [[2,1,1,0],[1,1,1],[1,1],[1]] => 3
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [[1,1,1],[1,1],[1]] => 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [[2,2,0,0],[2,1,0],[2,0],[1]] => 2
[1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0] => [3,1] => [[1,3,4],[2]] => [[3,1,0,0],[2,1,0],[1,1],[1]] => 3
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [[1,3],[2]] => [[2,1,0],[1,1],[1]] => 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [[1],[2]] => [[1,1],[1]] => 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => [[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [[1,2]] => [[2,0],[1]] => 1
[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]] => [[1,1,1,1],[1,1,1],[1,1],[1]] => 1
[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]] => [[2,1,1,0],[1,1,1],[1,1],[1]] => 3
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [[1,1,1],[1,1],[1]] => 1
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [[2,2,0,0],[2,1,0],[2,0],[1]] => 2
[1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0] => [3,1] => [[1,3,4],[2]] => [[3,1,0,0],[2,1,0],[1,1],[1]] => 3
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [[1,3],[2]] => [[2,1,0],[1,1],[1]] => 2
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [[1],[2]] => [[1,1],[1]] => 1
[1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0] => [4] => [[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => [[1,2]] => [[2,0],[1]] => 1
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => [[1],[2]] => [[1,1],[1]] => 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => [[1,2]] => [[2,0],[1]] => 1
[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]] => [[1,1],[1]] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => [[1,2]] => [[2,0],[1]] => 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [[1,1,1],[1,1],[1]] => 1
[1,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1,1] => [[1],[2],[3],[4]] => [[1,1,1,1],[1,1,1],[1,1],[1]] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,0] => [4] => [[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0] => [3] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [[2,2,0,0],[2,1,0],[2,0],[1]] => 2
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [[2,1,1,0],[1,1,1],[1,1],[1]] => 3
[1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0] => [3,1] => [[1,3,4],[2]] => [[3,1,0,0],[2,1,0],[1,1],[1]] => 3
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => [[1,3],[2]] => [[2,1,0],[1,1],[1]] => 2
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1] => [[1,3],[2]] => [[2,1,0],[1,1],[1]] => 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0] => [3] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0] => [3,1] => [[1,3,4],[2]] => [[3,1,0,0],[2,1,0],[1,1],[1]] => 3
[1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1,1] => [[1,4],[2],[3]] => [[2,1,1,0],[1,1,1],[1,1],[1]] => 3
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1] => [[1],[2],[3]] => [[1,1,1],[1,1],[1]] => 1
[1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0] => [2,2] => [[1,2],[3,4]] => [[2,2,0,0],[2,1,0],[2,0],[1]] => 2
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0] => [3] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => 1
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0] => [1,1] => [[1],[2]] => [[1,1],[1]] => 1
search for individual values
searching the database for the individual values of this statistic
Description
The order of promotion on a Gelfand-Tsetlin pattern.
Map
to Gelfand-Tsetlin pattern
Description
Sends a tableau to its corresponding Gelfand-Tsetlin pattern.
To obtain this Gelfand-Tsetlin pattern, fill in the first row of the pattern with the shape of the tableau.
Then remove the maximal entry from the tableau to obtain a smaller tableau, and repeat the process until the tableau is empty.
To obtain this Gelfand-Tsetlin pattern, fill in the first row of the pattern with the shape of the tableau.
Then remove the maximal entry from the tableau to obtain a smaller tableau, and repeat the process until the tableau is empty.
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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