Identifier
-
Mp00082:
Standard tableaux
—to Gelfand-Tsetlin pattern⟶
Gelfand-Tsetlin patterns
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
St000737: Semistandard tableaux ⟶ ℤ
Values
[[1]] => [[1]] => [[1]] => 1
[[1,2]] => [[2,0],[1]] => [[1,2]] => 1
[[1],[2]] => [[1,1],[1]] => [[1],[2]] => 1
[[1,2,3]] => [[3,0,0],[2,0],[1]] => [[1,2,3]] => 1
[[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]] => 1
[[1],[2],[3]] => [[1,1,1],[1,1],[1]] => [[1],[2],[3]] => 1
[[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,4]] => 1
[[1,3,4],[2]] => [[3,1,0,0],[2,1,0],[1,1],[1]] => [[1,3,4],[2]] => 1
[[1,2,4],[3]] => [[3,1,0,0],[2,1,0],[2,0],[1]] => [[1,2,4],[3]] => 1
[[1,2,3],[4]] => [[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3],[4]] => 1
[[1,3],[2,4]] => [[2,2,0,0],[2,1,0],[1,1],[1]] => [[1,3],[2,4]] => 4
[[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]] => 1
[[1,2],[3],[4]] => [[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2],[3],[4]] => 1
[[1],[2],[3],[4]] => [[1,1,1,1],[1,1,1],[1,1],[1]] => [[1],[2],[3],[4]] => 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
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 4
[[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]] => 5
[[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]] => 5
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 5
[[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]] => 5
[[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]] => 4
[[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]] => 4
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 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
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 4
[[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]] => 4
[[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]] => 5
[[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]] => 5
[[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]] => 5
[[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]] => 6
[[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]] => 6
[[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]] => 6
[[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]] => 6
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 4
[[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]] => 4
[[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]] => 5
[[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]] => 5
[[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]] => 5
[[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]] => 5
[[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]] => 5
[[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]] => 5
[[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]] => 4
[[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]] => 4
[[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]] => 6
[[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]] => 6
[[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]] => 6
[[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]] => 6
[[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]] => 6
[[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]] => 6
[[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]] => 4
[[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]] => 4
[[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]] => 5
[[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]] => 5
[[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]] => 5
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 1
[[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]] => 5
[[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]] => 5
>>> Load all 119 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The last entry on the main diagonal 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 $k=1,\dots,n$.
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.
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.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!