Identifier
Values
[[1]] => [[1]] => [[1]] => 0
[[1,2]] => [[2,0],[1]] => [[1,2]] => 1
[[1],[2]] => [[1,1],[1]] => [[1],[2]] => 0
[[1,2,3]] => [[3,0,0],[2,0],[1]] => [[1,2,3]] => 3
[[1,3],[2]] => [[2,1,0],[1,1],[1]] => [[1,3],[2]] => 1
[[1,2],[3]] => [[2,1,0],[2,0],[1]] => [[1,2],[3]] => 2
[[1],[2],[3]] => [[1,1,1],[1,1],[1]] => [[1],[2],[3]] => 0
[[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,4]] => 6
[[1,3,4],[2]] => [[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4],[2]] => 3
[[1,2,4],[3]] => [[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4],[3]] => 4
[[1,2,3],[4]] => [[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3],[4]] => 5
[[1,3],[2,4]] => [[2,2,0,0],[2,1,0],[1,1],[1]] => [[1,3],[2,4]] => 2
[[1,2],[3,4]] => [[2,2,0,0],[2,1,0],[2,0],[1]] => [[1,2],[3,4]] => 4
[[1,4],[2],[3]] => [[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4],[2],[3]] => 1
[[1,3],[2],[4]] => [[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3],[2],[4]] => 2
[[1,2],[3],[4]] => [[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2],[3],[4]] => 3
[[1],[2],[3],[4]] => [[1,1,1,1],[1,1,1],[1,1],[1]] => [[1],[2],[3],[4]] => 0
[[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]] => 10
[[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]] => 6
[[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]] => 7
[[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]] => 8
[[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]] => 9
[[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]] => 4
[[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]] => 7
[[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]] => 5
[[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]] => 6
[[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]] => 8
[[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
[[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
[[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]] => 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
[[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]] => 6
[[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]] => 7
[[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]] => 2
[[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
[[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]] => 5
[[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]] => 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]] => 6
[[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]] => 1
[[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]] => 2
[[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]] => 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]] => 4
[[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]] => 0
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Description
The charge of a semistandard tableau.
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
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.