Identifier
Values
[[1]] => [[1]] => [[1]] => 1
[[1,0],[0,1]] => [[1,1],[2]] => [[2,1],[2]] => 2
[[0,1],[1,0]] => [[1,2],[2]] => [[2,1],[1]] => 2
[[1,0,0],[0,1,0],[0,0,1]] => [[1,1,1],[2,2],[3]] => [[3,2,1],[3,2],[3]] => 3
[[0,1,0],[1,0,0],[0,0,1]] => [[1,1,2],[2,2],[3]] => [[3,2,1],[3,2],[2]] => 3
[[1,0,0],[0,0,1],[0,1,0]] => [[1,1,1],[2,3],[3]] => [[3,2,1],[3,1],[3]] => 3
[[0,1,0],[1,-1,1],[0,1,0]] => [[1,1,2],[2,3],[3]] => [[3,2,1],[3,1],[2]] => 4
[[0,0,1],[1,0,0],[0,1,0]] => [[1,1,3],[2,3],[3]] => [[3,2,1],[2,1],[2]] => 3
[[0,1,0],[0,0,1],[1,0,0]] => [[1,2,2],[2,3],[3]] => [[3,2,1],[3,1],[1]] => 3
[[0,0,1],[0,1,0],[1,0,0]] => [[1,2,3],[2,3],[3]] => [[3,2,1],[2,1],[1]] => 3
[[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]] => 4
[[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]] => 4
[[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]] => 4
[[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]] => 5
[[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]] => 4
[[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]] => 4
[[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]] => 4
[[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]] => 4
[[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]] => 4
[[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]] => 5
[[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]] => 6
[[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]] => 5
[[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]] => 5
[[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]] => 5
[[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]] => 4
[[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]] => 5
[[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]] => 5
[[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]] => 4
[[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]] => 4
[[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]] => 5
[[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]] => 4
[[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]] => 4
[[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]] => 5
[[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]] => 4
[[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]] => 5
[[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]] => 5
[[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]] => 4
[[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]] => 5
[[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]] => 5
[[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]] => 4
[[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]] => 5
[[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]] => 6
[[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]] => 5
[[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]] => 4
[[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]] => 4
[[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]] => 4
[[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]] => 4
[[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]] => 4
[[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]] => 5
[[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]] => 4
[[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]] => 4
[[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]] => 4
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Description
The total number of tiles in the Gelfand-Tsetlin pattern.
The tiling of a Gelfand-Tsetlin pattern is the finest partition of the entries in the pattern, such that adjacent (NW,NE,SW,SE) entries that are equal belong to the same part. These parts are called tiles, and each entry in a pattern belongs to exactly one tile.
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.