Identifier
Mp00082:
Standard tableaux
—to Gelfand-Tsetlin pattern⟶
Gelfand-Tsetlin patterns
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Images
[[1]] => [[1]] => [[1]] => [1]
[[1,2]] => [[2,0],[1]] => [[1,2]] => [1,2]
[[1],[2]] => [[1,1],[1]] => [[1],[2]] => [2,1]
[[1,2,3]] => [[3,0,0],[2,0],[1]] => [[1,2,3]] => [1,2,3]
[[1,3],[2]] => [[2,1,0],[1,1],[1]] => [[1,3],[2]] => [2,1,3]
[[1,2],[3]] => [[2,1,0],[2,0],[1]] => [[1,2],[3]] => [3,1,2]
[[1],[2],[3]] => [[1,1,1],[1,1],[1]] => [[1],[2],[3]] => [3,2,1]
[[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,4]] => [1,2,3,4]
[[1,3,4],[2]] => [[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4],[2]] => [2,1,3,4]
[[1,2,4],[3]] => [[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4],[3]] => [3,1,2,4]
[[1,2,3],[4]] => [[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3],[4]] => [4,1,2,3]
[[1,3],[2,4]] => [[2,2,0,0],[2,1,0],[1,1],[1]] => [[1,3],[2,4]] => [2,4,1,3]
[[1,2],[3,4]] => [[2,2,0,0],[2,1,0],[2,0],[1]] => [[1,2],[3,4]] => [3,4,1,2]
[[1,4],[2],[3]] => [[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4],[2],[3]] => [3,2,1,4]
[[1,3],[2],[4]] => [[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3],[2],[4]] => [4,2,1,3]
[[1,2],[3],[4]] => [[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2],[3],[4]] => [4,3,1,2]
[[1],[2],[3],[4]] => [[1,1,1,1],[1,1,1],[1,1],[1]] => [[1],[2],[3],[4]] => [4,3,2,1]
[[1,2,3,4,5]] => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,4,5]] => [1,2,3,4,5]
[[1,3,4,5],[2]] => [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4,5],[2]] => [2,1,3,4,5]
[[1,2,4,5],[3]] => [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4,5],[3]] => [3,1,2,4,5]
[[1,2,3,5],[4]] => [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,5],[4]] => [4,1,2,3,5]
[[1,2,3,4],[5]] => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,4],[5]] => [5,1,2,3,4]
[[1,3,5],[2,4]] => [[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]] => [[1,3,5],[2,4]] => [2,4,1,3,5]
[[1,2,5],[3,4]] => [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]] => [[1,2,5],[3,4]] => [3,4,1,2,5]
[[1,3,4],[2,5]] => [[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4],[2,5]] => [2,5,1,3,4]
[[1,2,4],[3,5]] => [[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4],[3,5]] => [3,5,1,2,4]
[[1,2,3],[4,5]] => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3],[4,5]] => [4,5,1,2,3]
[[1,4,5],[2],[3]] => [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4,5],[2],[3]] => [3,2,1,4,5]
[[1,3,5],[2],[4]] => [[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3,5],[2],[4]] => [4,2,1,3,5]
[[1,2,5],[3],[4]] => [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2,5],[3],[4]] => [4,3,1,2,5]
[[1,3,4],[2],[5]] => [[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4],[2],[5]] => [5,2,1,3,4]
[[1,2,4],[3],[5]] => [[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4],[3],[5]] => [5,3,1,2,4]
[[1,2,3],[4],[5]] => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3],[4],[5]] => [5,4,1,2,3]
[[1,4],[2,5],[3]] => [[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4],[2,5],[3]] => [3,2,5,1,4]
[[1,3],[2,5],[4]] => [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3],[2,5],[4]] => [4,2,5,1,3]
[[1,2],[3,5],[4]] => [[2,2,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2],[3,5],[4]] => [4,3,5,1,2]
[[1,3],[2,4],[5]] => [[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]] => [[1,3],[2,4],[5]] => [5,2,4,1,3]
[[1,2],[3,4],[5]] => [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]] => [[1,2],[3,4],[5]] => [5,3,4,1,2]
[[1,5],[2],[3],[4]] => [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]] => [[1,5],[2],[3],[4]] => [4,3,2,1,5]
[[1,4],[2],[3],[5]] => [[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4],[2],[3],[5]] => [5,3,2,1,4]
[[1,3],[2],[4],[5]] => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3],[2],[4],[5]] => [5,4,2,1,3]
[[1,2],[3],[4],[5]] => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2],[3],[4],[5]] => [5,4,3,1,2]
[[1],[2],[3],[4],[5]] => [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1]
[[1,2,3,4,5,6]] => [[6,0,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6]
[[1,3,4,5,6],[2]] => [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6]
[[1,2,4,5,6],[3]] => [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4,5,6],[3]] => [3,1,2,4,5,6]
[[1,2,3,5,6],[4]] => [[5,1,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,5,6],[4]] => [4,1,2,3,5,6]
[[1,2,3,4,6],[5]] => [[5,1,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,4,6],[5]] => [5,1,2,3,4,6]
[[1,2,3,4,5],[6]] => [[5,1,0,0,0,0],[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,4,5],[6]] => [6,1,2,3,4,5]
[[1,3,5,6],[2,4]] => [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]] => [[1,3,5,6],[2,4]] => [2,4,1,3,5,6]
[[1,2,5,6],[3,4]] => [[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6]
[[1,3,4,6],[2,5]] => [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4,6],[2,5]] => [2,5,1,3,4,6]
[[1,2,4,6],[3,5]] => [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4,6],[3,5]] => [3,5,1,2,4,6]
[[1,2,3,6],[4,5]] => [[4,2,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,6],[4,5]] => [4,5,1,2,3,6]
[[1,3,4,5],[2,6]] => [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4,5],[2,6]] => [2,6,1,3,4,5]
[[1,2,4,5],[3,6]] => [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4,5],[3,6]] => [3,6,1,2,4,5]
[[1,2,3,5],[4,6]] => [[4,2,0,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,5],[4,6]] => [4,6,1,2,3,5]
[[1,2,3,4],[5,6]] => [[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,4],[5,6]] => [5,6,1,2,3,4]
[[1,4,5,6],[2],[3]] => [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6]
[[1,3,5,6],[2],[4]] => [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3,5,6],[2],[4]] => [4,2,1,3,5,6]
[[1,2,5,6],[3],[4]] => [[4,1,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2,5,6],[3],[4]] => [4,3,1,2,5,6]
[[1,3,4,6],[2],[5]] => [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4,6],[2],[5]] => [5,2,1,3,4,6]
[[1,2,4,6],[3],[5]] => [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4,6],[3],[5]] => [5,3,1,2,4,6]
[[1,2,3,6],[4],[5]] => [[4,1,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,6],[4],[5]] => [5,4,1,2,3,6]
[[1,3,4,5],[2],[6]] => [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4,5],[2],[6]] => [6,2,1,3,4,5]
[[1,2,4,5],[3],[6]] => [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4,5],[3],[6]] => [6,3,1,2,4,5]
[[1,2,3,5],[4],[6]] => [[4,1,1,0,0,0],[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,5],[4],[6]] => [6,4,1,2,3,5]
[[1,2,3,4],[5],[6]] => [[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,4],[5],[6]] => [6,5,1,2,3,4]
[[1,3,5],[2,4,6]] => [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]] => [[1,3,5],[2,4,6]] => [2,4,6,1,3,5]
[[1,2,5],[3,4,6]] => [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]] => [[1,2,5],[3,4,6]] => [3,4,6,1,2,5]
[[1,3,4],[2,5,6]] => [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4],[2,5,6]] => [2,5,6,1,3,4]
[[1,2,4],[3,5,6]] => [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4],[3,5,6]] => [3,5,6,1,2,4]
[[1,2,3],[4,5,6]] => [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3]
[[1,4,6],[2,5],[3]] => [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4,6],[2,5],[3]] => [3,2,5,1,4,6]
[[1,3,6],[2,5],[4]] => [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6]
[[1,2,6],[3,5],[4]] => [[3,2,1,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2,6],[3,5],[4]] => [4,3,5,1,2,6]
[[1,3,6],[2,4],[5]] => [[3,2,1,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]] => [[1,3,6],[2,4],[5]] => [5,2,4,1,3,6]
[[1,2,6],[3,4],[5]] => [[3,2,1,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]] => [[1,2,6],[3,4],[5]] => [5,3,4,1,2,6]
[[1,4,5],[2,6],[3]] => [[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4,5],[2,6],[3]] => [3,2,6,1,4,5]
[[1,3,5],[2,6],[4]] => [[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3,5],[2,6],[4]] => [4,2,6,1,3,5]
[[1,2,5],[3,6],[4]] => [[3,2,1,0,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2,5],[3,6],[4]] => [4,3,6,1,2,5]
[[1,3,4],[2,6],[5]] => [[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4],[2,6],[5]] => [5,2,6,1,3,4]
[[1,2,4],[3,6],[5]] => [[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4],[3,6],[5]] => [5,3,6,1,2,4]
[[1,2,3],[4,6],[5]] => [[3,2,1,0,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3],[4,6],[5]] => [5,4,6,1,2,3]
[[1,3,5],[2,4],[6]] => [[3,2,1,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]] => [[1,3,5],[2,4],[6]] => [6,2,4,1,3,5]
[[1,2,5],[3,4],[6]] => [[3,2,1,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]] => [[1,2,5],[3,4],[6]] => [6,3,4,1,2,5]
[[1,3,4],[2,5],[6]] => [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4],[2,5],[6]] => [6,2,5,1,3,4]
[[1,2,4],[3,5],[6]] => [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4],[3,5],[6]] => [6,3,5,1,2,4]
[[1,2,3],[4,5],[6]] => [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3],[4,5],[6]] => [6,4,5,1,2,3]
[[1,5,6],[2],[3],[4]] => [[3,1,1,1,0,0],[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6]
[[1,4,6],[2],[3],[5]] => [[3,1,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6]
[[1,3,6],[2],[4],[5]] => [[3,1,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6]
[[1,2,6],[3],[4],[5]] => [[3,1,1,1,0,0],[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6]
[[1,4,5],[2],[3],[6]] => [[3,1,1,1,0,0],[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5]
[[1,3,5],[2],[4],[6]] => [[3,1,1,1,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5]
[[1,2,5],[3],[4],[6]] => [[3,1,1,1,0,0],[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5]
[[1,3,4],[2],[5],[6]] => [[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4]
[[1,2,4],[3],[5],[6]] => [[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4]
[[1,2,3],[4],[5],[6]] => [[3,1,1,1,0,0],[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3],[4],[5],[6]] => [6,5,4,1,2,3]
[[1,4],[2,5],[3,6]] => [[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4],[2,5],[3,6]] => [3,6,2,5,1,4]
[[1,3],[2,5],[4,6]] => [[2,2,2,0,0,0],[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3],[2,5],[4,6]] => [4,6,2,5,1,3]
>>> Load all 119 entries. <<<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
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 $k=1,\dots,n$.
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
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).
searching the database
Sorry, this map was not found in the database.