Identifier
-
Mp00001:
Alternating sign matrices
—to semistandard tableau via monotone triangles⟶
Semistandard tableaux
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤ
Values
[[1]] => [[1]] => [[1]] => 0
[[1,0],[0,1]] => [[1,1],[2]] => [[2,1],[2]] => 0
[[0,1],[1,0]] => [[1,2],[2]] => [[2,1],[1]] => 0
[[1,0,0],[0,1,0],[0,0,1]] => [[1,1,1],[2,2],[3]] => [[3,2,1],[3,2],[3]] => 0
[[0,1,0],[1,0,0],[0,0,1]] => [[1,1,2],[2,2],[3]] => [[3,2,1],[3,2],[2]] => 0
[[1,0,0],[0,0,1],[0,1,0]] => [[1,1,1],[2,3],[3]] => [[3,2,1],[3,1],[3]] => 0
[[0,1,0],[1,-1,1],[0,1,0]] => [[1,1,2],[2,3],[3]] => [[3,2,1],[3,1],[2]] => 1
[[0,0,1],[1,0,0],[0,1,0]] => [[1,1,3],[2,3],[3]] => [[3,2,1],[2,1],[2]] => 0
[[0,1,0],[0,0,1],[1,0,0]] => [[1,2,2],[2,3],[3]] => [[3,2,1],[3,1],[1]] => 0
[[0,0,1],[0,1,0],[1,0,0]] => [[1,2,3],[2,3],[3]] => [[3,2,1],[2,1],[1]] => 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => [[1,1,1,1],[2,2,2],[3,3],[4]] => [[4,3,2,1],[4,3,2],[4,3],[4]] => 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => [[1,1,1,2],[2,2,2],[3,3],[4]] => [[4,3,2,1],[4,3,2],[4,3],[3]] => 0
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,1],[2,2,3],[3,3],[4]] => [[4,3,2,1],[4,3,2],[4,2],[4]] => 0
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,2],[2,2,3],[3,3],[4]] => [[4,3,2,1],[4,3,2],[4,2],[3]] => 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => [[1,1,1,3],[2,2,3],[3,3],[4]] => [[4,3,2,1],[4,3,2],[3,2],[3]] => 0
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]] => [[1,1,2,2],[2,2,3],[3,3],[4]] => [[4,3,2,1],[4,3,2],[4,2],[2]] => 0
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => [[1,1,2,3],[2,2,3],[3,3],[4]] => [[4,3,2,1],[4,3,2],[3,2],[2]] => 0
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => [[1,1,1,1],[2,2,2],[3,4],[4]] => [[4,3,2,1],[4,3,1],[4,3],[4]] => 0
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => [[1,1,1,2],[2,2,2],[3,4],[4]] => [[4,3,2,1],[4,3,1],[4,3],[3]] => 0
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => [[1,1,1,1],[2,2,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[4,2],[4]] => 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]] => [[1,1,1,2],[2,2,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[4,2],[3]] => 2
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]] => [[1,1,1,3],[2,2,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[3,2],[3]] => 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]] => [[1,1,2,2],[2,2,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[4,2],[2]] => 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]] => [[1,1,2,3],[2,2,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[3,2],[2]] => 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => [[1,1,1,1],[2,2,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[4,2],[4]] => 0
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]] => [[1,1,1,2],[2,2,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[4,2],[3]] => 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]] => [[1,1,1,3],[2,2,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[3,2],[3]] => 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => [[1,1,1,4],[2,2,4],[3,4],[4]] => [[4,3,2,1],[3,2,1],[3,2],[3]] => 0
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]] => [[1,1,2,2],[2,2,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[4,2],[2]] => 0
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]] => [[1,1,2,3],[2,2,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[3,2],[2]] => 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]] => [[1,1,2,4],[2,2,4],[3,4],[4]] => [[4,3,2,1],[3,2,1],[3,2],[2]] => 0
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => [[1,1,1,1],[2,3,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[4,1],[4]] => 0
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]] => [[1,1,1,2],[2,3,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[4,1],[3]] => 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]] => [[1,1,1,3],[2,3,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[3,1],[3]] => 0
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]] => [[1,1,2,2],[2,3,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[4,1],[2]] => 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]] => [[1,1,2,3],[2,3,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[3,1],[2]] => 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,1],[2,3,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[4,1],[4]] => 0
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,2],[2,3,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[4,1],[3]] => 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]] => [[1,1,1,3],[2,3,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[3,1],[3]] => 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => [[1,1,1,4],[2,3,4],[3,4],[4]] => [[4,3,2,1],[3,2,1],[3,1],[3]] => 0
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]] => [[1,1,2,2],[2,3,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[4,1],[2]] => 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]] => [[1,1,2,3],[2,3,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[3,1],[2]] => 2
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]] => [[1,1,2,4],[2,3,4],[3,4],[4]] => [[4,3,2,1],[3,2,1],[3,1],[2]] => 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]] => [[1,1,3,3],[2,3,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[2,1],[2]] => 0
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]] => [[1,1,3,4],[2,3,4],[3,4],[4]] => [[4,3,2,1],[3,2,1],[2,1],[2]] => 0
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]] => [[1,2,2,2],[2,3,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[4,1],[1]] => 0
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]] => [[1,2,2,3],[2,3,3],[3,4],[4]] => [[4,3,2,1],[4,3,1],[3,1],[1]] => 0
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]] => [[1,2,2,2],[2,3,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[4,1],[1]] => 0
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]] => [[1,2,2,3],[2,3,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[3,1],[1]] => 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]] => [[1,2,2,4],[2,3,4],[3,4],[4]] => [[4,3,2,1],[3,2,1],[3,1],[1]] => 0
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]] => [[1,2,3,3],[2,3,4],[3,4],[4]] => [[4,3,2,1],[4,2,1],[2,1],[1]] => 0
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => [[1,2,3,4],[2,3,4],[3,4],[4]] => [[4,3,2,1],[3,2,1],[2,1],[1]] => 0
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Description
The number of special entries.
An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
Map
to Gelfand-Tsetlin pattern
Description
Return the Gelfand-Tsetlin pattern corresponding to the semistandard tableau.
Map
to semistandard tableau via monotone triangles
Description
The semistandard tableau corresponding the monotone triangle of an alternating sign matrix.
This is obtained by interpreting each row of the monotone triangle as an integer partition, and filling the cells of the smallest partition with ones, the second smallest with twos, and so on.
This is obtained by interpreting each row of the monotone triangle as an integer partition, and filling the cells of the smallest partition with ones, the second smallest with twos, and so on.
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