Identifier
-
Mp00042:
Integer partitions
—initial tableau⟶
Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
St000736: Semistandard tableaux ⟶ ℤ
Values
[1] => [[1]] => [[1]] => [[1]] => 1
[2] => [[1,2]] => [[2,0],[1]] => [[1,2]] => 2
[1,1] => [[1],[2]] => [[1,1],[1]] => [[1],[2]] => 1
[3] => [[1,2,3]] => [[3,0,0],[2,0],[1]] => [[1,2,3]] => 3
[2,1] => [[1,2],[3]] => [[2,1,0],[2,0],[1]] => [[1,2],[3]] => 2
[1,1,1] => [[1],[2],[3]] => [[1,1,1],[1,1],[1]] => [[1],[2],[3]] => 1
[4] => [[1,2,3,4]] => [[4,0,0,0],[3,0,0],[2,0],[1]] => [[1,2,3,4]] => 4
[3,1] => [[1,2,3],[4]] => [[3,1,0,0],[3,0,0],[2,0],[1]] => [[1,2,3],[4]] => 3
[2,2] => [[1,2],[3,4]] => [[2,2,0,0],[2,1,0],[2,0],[1]] => [[1,2],[3,4]] => 2
[2,1,1] => [[1,2],[3],[4]] => [[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2],[3],[4]] => 2
[1,1,1,1] => [[1],[2],[3],[4]] => [[1,1,1,1],[1,1,1],[1,1],[1]] => [[1],[2],[3],[4]] => 1
[5] => [[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]] => 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]] => 4
[3,2] => [[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]] => 3
[3,1,1] => [[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]] => 3
[2,2,1] => [[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]] => 2
[2,1,1,1] => [[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]] => 2
[1,1,1,1,1] => [[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
[6] => [[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]] => 6
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Description
The last entry in the first row of a semistandard tableau.
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 λ(k) be its (n−k+1)-st row. The defining inequalities of a Gelfand-Tsetlin pattern imply, regarding each row as a partition,
λ(0)⊆λ(1)⊆⋯⊆λ(n),
where λ(0) is the empty partition.
Each skew shape λ(k)/λ(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 λ(k)/λ(k−1), for k=1,…,n.
Let G be a Gelfand-Tsetlin pattern and let λ(k) be its (n−k+1)-st row. The defining inequalities of a Gelfand-Tsetlin pattern imply, regarding each row as a partition,
λ(0)⊆λ(1)⊆⋯⊆λ(n),
where λ(0) is the empty partition.
Each skew shape λ(k)/λ(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 λ(k)/λ(k−1), for k=1,…,n.
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers 1 through n row by row.
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