- FindStat

Identifier

Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000228: Integer partitions ⟶ ℤ

Values

[[1,0],[0]] => [[2]] => [1] => 1

[[1,0],[1]] => [[1]] => [1] => 1

[[2,0],[0]] => [[2,2]] => [2] => 2

[[2,0],[1]] => [[1,2]] => [2] => 2

[[2,0],[2]] => [[1,1]] => [2] => 2

[[1,1],[1]] => [[1],[2]] => [1,1] => 2

[[1,0,0],[0,0],[0]] => [[3]] => [1] => 1

[[1,0,0],[1,0],[0]] => [[2]] => [1] => 1

[[1,0,0],[1,0],[1]] => [[1]] => [1] => 1

[[3,0],[0]] => [[2,2,2]] => [3] => 3

[[3,0],[1]] => [[1,2,2]] => [3] => 3

[[3,0],[2]] => [[1,1,2]] => [3] => 3

[[3,0],[3]] => [[1,1,1]] => [3] => 3

[[2,1],[1]] => [[1,2],[2]] => [2,1] => 3

[[2,1],[2]] => [[1,1],[2]] => [2,1] => 3

[[2,0,0],[0,0],[0]] => [[3,3]] => [2] => 2

[[2,0,0],[1,0],[0]] => [[2,3]] => [2] => 2

[[2,0,0],[1,0],[1]] => [[1,3]] => [2] => 2

[[2,0,0],[2,0],[0]] => [[2,2]] => [2] => 2

[[2,0,0],[2,0],[1]] => [[1,2]] => [2] => 2

[[2,0,0],[2,0],[2]] => [[1,1]] => [2] => 2

[[1,1,0],[1,0],[0]] => [[2],[3]] => [1,1] => 2

[[1,1,0],[1,0],[1]] => [[1],[3]] => [1,1] => 2

[[1,1,0],[1,1],[1]] => [[1],[2]] => [1,1] => 2

[[1,0,0,0],[0,0,0],[0,0],[0]] => [[4]] => [1] => 1

[[1,0,0,0],[1,0,0],[0,0],[0]] => [[3]] => [1] => 1

[[1,0,0,0],[1,0,0],[1,0],[0]] => [[2]] => [1] => 1

[[1,0,0,0],[1,0,0],[1,0],[1]] => [[1]] => [1] => 1

[[4,0],[0]] => [[2,2,2,2]] => [4] => 4

[[4,0],[1]] => [[1,2,2,2]] => [4] => 4

[[4,0],[2]] => [[1,1,2,2]] => [4] => 4

[[4,0],[3]] => [[1,1,1,2]] => [4] => 4

[[4,0],[4]] => [[1,1,1,1]] => [4] => 4

[[3,1],[1]] => [[1,2,2],[2]] => [3,1] => 4

[[3,1],[2]] => [[1,1,2],[2]] => [3,1] => 4

[[3,1],[3]] => [[1,1,1],[2]] => [3,1] => 4

[[2,2],[2]] => [[1,1],[2,2]] => [2,2] => 4

[[3,0,0],[0,0],[0]] => [[3,3,3]] => [3] => 3

[[3,0,0],[1,0],[0]] => [[2,3,3]] => [3] => 3

[[3,0,0],[1,0],[1]] => [[1,3,3]] => [3] => 3

[[3,0,0],[2,0],[0]] => [[2,2,3]] => [3] => 3

[[3,0,0],[2,0],[1]] => [[1,2,3]] => [3] => 3

[[3,0,0],[2,0],[2]] => [[1,1,3]] => [3] => 3

[[3,0,0],[3,0],[0]] => [[2,2,2]] => [3] => 3

[[3,0,0],[3,0],[1]] => [[1,2,2]] => [3] => 3

[[3,0,0],[3,0],[2]] => [[1,1,2]] => [3] => 3

[[3,0,0],[3,0],[3]] => [[1,1,1]] => [3] => 3

[[2,1,0],[1,0],[0]] => [[2,3],[3]] => [2,1] => 3

[[2,1,0],[1,0],[1]] => [[1,3],[3]] => [2,1] => 3

[[2,1,0],[1,1],[1]] => [[1,3],[2]] => [2,1] => 3

[[2,1,0],[2,0],[0]] => [[2,2],[3]] => [2,1] => 3

[[2,1,0],[2,0],[1]] => [[1,2],[3]] => [2,1] => 3

[[2,1,0],[2,0],[2]] => [[1,1],[3]] => [2,1] => 3

[[2,1,0],[2,1],[1]] => [[1,2],[2]] => [2,1] => 3

[[2,1,0],[2,1],[2]] => [[1,1],[2]] => [2,1] => 3

[[1,1,1],[1,1],[1]] => [[1],[2],[3]] => [1,1,1] => 3

[[2,0,0,0],[0,0,0],[0,0],[0]] => [[4,4]] => [2] => 2

[[2,0,0,0],[1,0,0],[0,0],[0]] => [[3,4]] => [2] => 2

[[2,0,0,0],[1,0,0],[1,0],[0]] => [[2,4]] => [2] => 2

[[2,0,0,0],[1,0,0],[1,0],[1]] => [[1,4]] => [2] => 2

[[2,0,0,0],[2,0,0],[0,0],[0]] => [[3,3]] => [2] => 2

[[2,0,0,0],[2,0,0],[1,0],[0]] => [[2,3]] => [2] => 2

[[2,0,0,0],[2,0,0],[1,0],[1]] => [[1,3]] => [2] => 2

[[2,0,0,0],[2,0,0],[2,0],[0]] => [[2,2]] => [2] => 2

[[2,0,0,0],[2,0,0],[2,0],[1]] => [[1,2]] => [2] => 2

[[2,0,0,0],[2,0,0],[2,0],[2]] => [[1,1]] => [2] => 2

[[1,1,0,0],[1,0,0],[0,0],[0]] => [[3],[4]] => [1,1] => 2

[[1,1,0,0],[1,0,0],[1,0],[0]] => [[2],[4]] => [1,1] => 2

[[1,1,0,0],[1,0,0],[1,0],[1]] => [[1],[4]] => [1,1] => 2

[[1,1,0,0],[1,1,0],[1,0],[0]] => [[2],[3]] => [1,1] => 2

[[1,1,0,0],[1,1,0],[1,0],[1]] => [[1],[3]] => [1,1] => 2

[[1,1,0,0],[1,1,0],[1,1],[1]] => [[1],[2]] => [1,1] => 2

[[1,0,0,0,0],[0,0,0,0],[0,0,0],[0,0],[0]] => [[5]] => [1] => 1

[[1,0,0,0,0],[1,0,0,0],[0,0,0],[0,0],[0]] => [[4]] => [1] => 1

[[1,0,0,0,0],[1,0,0,0],[1,0,0],[0,0],[0]] => [[3]] => [1] => 1

[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[0]] => [[2]] => [1] => 1

[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]] => [[1]] => [1] => 1

[[5,0],[0]] => [[2,2,2,2,2]] => [5] => 5

[[5,0],[1]] => [[1,2,2,2,2]] => [5] => 5

[[5,0],[2]] => [[1,1,2,2,2]] => [5] => 5

[[5,0],[3]] => [[1,1,1,2,2]] => [5] => 5

[[5,0],[4]] => [[1,1,1,1,2]] => [5] => 5

[[5,0],[5]] => [[1,1,1,1,1]] => [5] => 5

[[4,1],[1]] => [[1,2,2,2],[2]] => [4,1] => 5

[[4,1],[2]] => [[1,1,2,2],[2]] => [4,1] => 5

[[4,1],[3]] => [[1,1,1,2],[2]] => [4,1] => 5

[[4,1],[4]] => [[1,1,1,1],[2]] => [4,1] => 5

[[3,2],[2]] => [[1,1,2],[2,2]] => [3,2] => 5

[[3,2],[3]] => [[1,1,1],[2,2]] => [3,2] => 5

[[4,0,0],[0,0],[0]] => [[3,3,3,3]] => [4] => 4

[[4,0,0],[1,0],[0]] => [[2,3,3,3]] => [4] => 4

[[4,0,0],[1,0],[1]] => [[1,3,3,3]] => [4] => 4

[[4,0,0],[2,0],[0]] => [[2,2,3,3]] => [4] => 4

[[4,0,0],[2,0],[1]] => [[1,2,3,3]] => [4] => 4

[[4,0,0],[2,0],[2]] => [[1,1,3,3]] => [4] => 4

[[4,0,0],[3,0],[0]] => [[2,2,2,3]] => [4] => 4

[[4,0,0],[3,0],[1]] => [[1,2,2,3]] => [4] => 4

[[4,0,0],[3,0],[2]] => [[1,1,2,3]] => [4] => 4

[[4,0,0],[3,0],[3]] => [[1,1,1,3]] => [4] => 4

[[4,0,0],[4,0],[0]] => [[2,2,2,2]] => [4] => 4

[[4,0,0],[4,0],[1]] => [[1,2,2,2]] => [4] => 4

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Description

The size of a partition.
This statistic is the constant statistic of the level sets.

Map

to semistandard tableau

Description

Return the Gelfand-Tsetlin pattern as a semistandard Young tableau.
Let $G$ be a Gelfand-Tsetlin pattern and let $\lambda^{(k)}$ be its $(n-k+1)$-st row. The defining inequalities of a Gelfand-Tsetlin pattern imply, regarding each row as a partition,
$$ \lambda^{(0)} \subseteq \lambda^{(1)} \subseteq \cdots \subseteq \lambda^{(n)}, $$
where $\lambda^{(0)}$ is the empty partition.
Each skew shape $\lambda^{(k)} / \lambda^{(k-1)}$ is moreover a horizontal strip.
We now define a semistandard tableau $T(G)$ by inserting $k$ into the cells of the skew shape $\lambda^{(k)} / \lambda^{(k-1)}$, for $k=1,\dots,n$.

Map

shape

Description

Return the shape of a tableau.