Identifier
-
Mp00284:
Standard tableaux
—rows⟶
Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤ
Values
[[1]] => {{1}} => [1] => [[1]] => 1
[[1,2]] => {{1,2}} => [2,1] => [[0,1],[1,0]] => 2
[[1],[2]] => {{1},{2}} => [1,2] => [[1,0],[0,1]] => 1
[[1,2,3]] => {{1,2,3}} => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]] => 3
[[1,3],[2]] => {{1,3},{2}} => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]] => 3
[[1,2],[3]] => {{1,2},{3}} => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]] => 2
[[1],[2],[3]] => {{1},{2},{3}} => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]] => 1
[[1,2,3,4]] => {{1,2,3,4}} => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => 4
[[1,3,4],[2]] => {{1,3,4},{2}} => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]] => 4
[[1,2,4],[3]] => {{1,2,4},{3}} => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => 4
[[1,2,3],[4]] => {{1,2,3},{4}} => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => 3
[[1,3],[2,4]] => {{1,3},{2,4}} => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]] => 3
[[1,2],[3,4]] => {{1,2},{3,4}} => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => 2
[[1,4],[2],[3]] => {{1,4},{2},{3}} => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]] => 4
[[1,3],[2],[4]] => {{1,3},{2},{4}} => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => 3
[[1,2],[3],[4]] => {{1,2},{3},{4}} => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => 2
[[1],[2],[3],[4]] => {{1},{2},{3},{4}} => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => 1
[[1,2,3,4,5]] => {{1,2,3,4,5}} => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]] => 5
[[1,3,4,5],[2]] => {{1,3,4,5},{2}} => [3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]] => 5
[[1,2,4,5],[3]] => {{1,2,4,5},{3}} => [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]] => 5
[[1,2,3,5],[4]] => {{1,2,3,5},{4}} => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]] => 5
[[1,2,3,4],[5]] => {{1,2,3,4},{5}} => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 4
[[1,3,5],[2,4]] => {{1,3,5},{2,4}} => [3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]] => 5
[[1,2,5],[3,4]] => {{1,2,5},{3,4}} => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]] => 5
[[1,3,4],[2,5]] => {{1,3,4},{2,5}} => [3,5,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]] => 4
[[1,2,4],[3,5]] => {{1,2,4},{3,5}} => [2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]] => 4
[[1,2,3],[4,5]] => {{1,2,3},{4,5}} => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 3
[[1,4,5],[2],[3]] => {{1,4,5},{2},{3}} => [4,2,3,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]] => 5
[[1,3,5],[2],[4]] => {{1,3,5},{2},{4}} => [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]] => 5
[[1,2,5],[3],[4]] => {{1,2,5},{3},{4}} => [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]] => 5
[[1,3,4],[2],[5]] => {{1,3,4},{2},{5}} => [3,2,4,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 4
[[1,2,4],[3],[5]] => {{1,2,4},{3},{5}} => [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 4
[[1,2,3],[4],[5]] => {{1,2,3},{4},{5}} => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 3
[[1,4],[2,5],[3]] => {{1,4},{2,5},{3}} => [4,5,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]] => 4
[[1,3],[2,5],[4]] => {{1,3},{2,5},{4}} => [3,5,1,4,2] => [[0,0,1,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]] => 3
[[1,2],[3,5],[4]] => {{1,2},{3,5},{4}} => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]] => 2
[[1,3],[2,4],[5]] => {{1,3},{2,4},{5}} => [3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 3
[[1,2],[3,4],[5]] => {{1,2},{3,4},{5}} => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 2
[[1,5],[2],[3],[4]] => {{1,5},{2},{3},{4}} => [5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]] => 5
[[1,4],[2],[3],[5]] => {{1,4},{2},{3},{5}} => [4,2,3,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 4
[[1,3],[2],[4],[5]] => {{1,3},{2},{4},{5}} => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 3
[[1,2],[3],[4],[5]] => {{1,2},{3},{4},{5}} => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 2
[[1],[2],[3],[4],[5]] => {{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 1
[[1,2,3,4,5,6]] => {{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]] => 6
[[1,3,4,5,6],[2]] => {{1,3,4,5,6},{2}} => [3,2,4,5,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]] => 6
[[1,2,4,5,6],[3]] => {{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]] => 6
[[1,2,3,5,6],[4]] => {{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]] => 6
[[1,2,3,4,6],[5]] => {{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]] => 6
[[1,2,3,4,5],[6]] => {{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 5
[[1,3,5,6],[2,4]] => {{1,3,5,6},{2,4}} => [3,4,5,2,6,1] => [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]] => 6
[[1,2,5,6],[3,4]] => {{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]] => 6
[[1,3,4,6],[2,5]] => {{1,3,4,6},{2,5}} => [3,5,4,6,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]] => 6
[[1,2,4,6],[3,5]] => {{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]] => 6
[[1,2,3,6],[4,5]] => {{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]] => 6
[[1,2,3,5],[4,6]] => {{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]] => 5
[[1,2,3,4],[5,6]] => {{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 4
[[1,4,5,6],[2],[3]] => {{1,4,5,6},{2},{3}} => [4,2,3,5,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]] => 6
[[1,3,5,6],[2],[4]] => {{1,3,5,6},{2},{4}} => [3,2,5,4,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]] => 6
[[1,2,5,6],[3],[4]] => {{1,2,5,6},{3},{4}} => [2,5,3,4,6,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]] => 6
[[1,3,4,6],[2],[5]] => {{1,3,4,6},{2},{5}} => [3,2,4,6,5,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]] => 6
[[1,2,4,6],[3],[5]] => {{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]] => 6
[[1,2,3,6],[4],[5]] => {{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]] => 6
[[1,3,4,5],[2],[6]] => {{1,3,4,5},{2},{6}} => [3,2,4,5,1,6] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 5
[[1,2,4,5],[3],[6]] => {{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 5
[[1,2,3,5],[4],[6]] => {{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 5
[[1,2,3,4],[5],[6]] => {{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 4
[[1,3,5],[2,4,6]] => {{1,3,5},{2,4,6}} => [3,4,5,6,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]] => 5
[[1,2,4],[3,5,6]] => {{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]] => 4
[[1,2,3],[4,5,6]] => {{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]] => 3
[[1,4,6],[2,5],[3]] => {{1,4,6},{2,5},{3}} => [4,5,3,6,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]] => 6
[[1,3,6],[2,5],[4]] => {{1,3,6},{2,5},{4}} => [3,5,6,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]] => 6
[[1,2,6],[3,5],[4]] => {{1,2,6},{3,5},{4}} => [2,6,5,4,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]] => 6
[[1,2,6],[3,4],[5]] => {{1,2,6},{3,4},{5}} => [2,6,4,3,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]] => 6
[[1,4,5],[2,6],[3]] => {{1,4,5},{2,6},{3}} => [4,6,3,5,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0]] => 5
[[1,3,5],[2,6],[4]] => {{1,3,5},{2,6},{4}} => [3,6,5,4,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]] => 5
[[1,2,3],[4,6],[5]] => {{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]] => 3
[[1,3,5],[2,4],[6]] => {{1,3,5},{2,4},{6}} => [3,4,5,2,1,6] => [[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 5
[[1,2,5],[3,4],[6]] => {{1,2,5},{3,4},{6}} => [2,5,4,3,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]] => 5
[[1,2,4],[3,5],[6]] => {{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 4
[[1,2,3],[4,5],[6]] => {{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 3
[[1,5,6],[2],[3],[4]] => {{1,5,6},{2},{3},{4}} => [5,2,3,4,6,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0]] => 6
[[1,4,6],[2],[3],[5]] => {{1,4,6},{2},{3},{5}} => [4,2,3,6,5,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]] => 6
[[1,3,6],[2],[4],[5]] => {{1,3,6},{2},{4},{5}} => [3,2,6,4,5,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]] => 6
[[1,2,6],[3],[4],[5]] => {{1,2,6},{3},{4},{5}} => [2,6,3,4,5,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]] => 6
[[1,4,5],[2],[3],[6]] => {{1,4,5},{2},{3},{6}} => [4,2,3,5,1,6] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 5
[[1,3,5],[2],[4],[6]] => {{1,3,5},{2},{4},{6}} => [3,2,5,4,1,6] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 5
[[1,2,5],[3],[4],[6]] => {{1,2,5},{3},{4},{6}} => [2,5,3,4,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]] => 5
[[1,3,4],[2],[5],[6]] => {{1,3,4},{2},{5},{6}} => [3,2,4,1,5,6] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 4
[[1,2,4],[3],[5],[6]] => {{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 4
[[1,2,3],[4],[5],[6]] => {{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 3
[[1,4],[2,5],[3,6]] => {{1,4},{2,5},{3,6}} => [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]] => 4
[[1,3],[2,5],[4,6]] => {{1,3},{2,5},{4,6}} => [3,5,1,6,2,4] => [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,1,0,0]] => 3
[[1,2],[3,5],[4,6]] => {{1,2},{3,5},{4,6}} => [2,1,5,6,3,4] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0]] => 2
[[1,3],[2,4],[5,6]] => {{1,3},{2,4},{5,6}} => [3,4,1,2,6,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 3
[[1,2],[3,4],[5,6]] => {{1,2},{3,4},{5,6}} => [2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 2
[[1,5],[2,6],[3],[4]] => {{1,5},{2,6},{3},{4}} => [5,6,3,4,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]] => 5
[[1,2],[3,6],[4],[5]] => {{1,2},{3,6},{4},{5}} => [2,1,6,4,5,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0]] => 2
[[1,4],[2,5],[3],[6]] => {{1,4},{2,5},{3},{6}} => [4,5,3,1,2,6] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]] => 4
[[1,2],[3,5],[4],[6]] => {{1,2},{3,5},{4},{6}} => [2,1,5,4,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 2
[[1,3],[2,4],[5],[6]] => {{1,3},{2,4},{5},{6}} => [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 3
[[1,2],[3,4],[5],[6]] => {{1,2},{3,4},{5},{6}} => [2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 2
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Description
The column of the unique '1' in the first row of the alternating sign matrix.
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
Map
rows
Description
The set partition whose blocks are the rows of the tableau.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
to alternating sign matrix
Description
Maps a permutation to its permutation matrix as an alternating sign matrix.
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