Identifier
Values
[1] => [1,0,1,0] => [3,1,2] => {{1,3},{2}} => 1
[2] => [1,1,0,0,1,0] => [2,4,1,3] => {{1,2,4},{3}} => 2
[1,1] => [1,0,1,1,0,0] => [3,1,4,2] => {{1,3,4},{2}} => 1
[3] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => {{1,2,3,5},{4}} => 1
[2,1] => [1,0,1,0,1,0] => [4,1,2,3] => {{1,4},{2},{3}} => 2
[1,1,1] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => {{1,3,4,5},{2}} => 1
[4] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => {{1,2,3,4,6},{5}} => 2
[3,1] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => {{1,5},{2,3},{4}} => 1
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => {{1,2,4,5},{3}} => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => {{1,5},{2},{3,4}} => 2
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => {{1,3,4,5,6},{2}} => 1
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => {{1,2,3,4,5,7},{6}} => 1
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => {{1,6},{2,3,4},{5}} => 2
[3,2] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => {{1,2,5},{3},{4}} => 2
[3,1,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => {{1,3,5},{2},{4}} => 1
[2,2,1] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => {{1,4,5},{2},{3}} => 2
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}} => 2
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => {{1,3,4,5,6,7},{2}} => 1
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [7,3,4,5,1,2,6] => {{1,7},{2,3,4,5},{6}} => 1
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => {{1,2,6},{3,4},{5}} => 3
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => {{1,4,6},{2,3},{5}} => 2
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => {{1,2,3,5,6},{4}} => 1
[3,2,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => {{1,5},{2},{3},{4}} => 2
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}} => 1
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => {{1,2,4,5,6},{3}} => 2
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}} => 2
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => {{1,7},{2},{3,4,5,6}} => 2
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => {{1,2,7},{3,4,5},{6}} => 2
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [5,3,4,1,7,2,6] => {{1,5,7},{2,3,4},{6}} => 1
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => {{1,2,3,6},{4},{5}} => 2
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => {{1,6},{2,4},{3},{5}} => 3
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}} => 2
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}} => 1
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}} => 2
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}} => 2
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => {{1,3,7},{2},{4,5,6}} => 1
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}} => 2
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => {{1,6,7},{2},{3,4,5}} => 2
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => {{1,2,3,7},{4,5},{6}} => 1
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => {{1,7},{2,5},{3,4},{6}} => 2
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [4,3,1,5,7,2,6] => {{1,4,5,7},{2,3},{6}} => 1
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => {{1,2,3,4,6,7},{5}} => 2
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => {{1,6},{2,3},{4},{5}} => 2
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}} => 3
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}} => 3
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [3,1,4,7,6,2,5] => {{1,3,4,7},{2},{5,6}} => 2
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}} => 2
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}} => 1
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}} => 2
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => {{1,7},{2},{3,6},{4,5}} => 2
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => {{1,2,4,5,6,7},{3}} => 2
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => {{1,5,6,7},{2},{3,4}} => 2
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => {{1,2,3,4,7},{5},{6}} => 2
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [7,3,5,1,2,4,6] => {{1,7},{2,3,5},{4},{6}} => 1
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [2,5,4,1,7,3,6] => {{1,2,5,7},{3,4},{6}} => 2
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,1,5,2,3,6] => {{1,7},{2,4,5},{3},{6}} => 2
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [3,1,4,5,7,2,6] => {{1,3,4,5,7},{2},{6}} => 1
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [6,3,4,1,2,7,5] => {{1,6,7},{2,3,4},{5}} => 2
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => {{1,2,6},{3},{4},{5}} => 3
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}} => 2
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => {{1,4,6},{2},{3},{5}} => 3
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => {{1,7},{2},{3,4,6},{5}} => 3
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => {{1,2,3,5,6,7},{4}} => 1
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}} => 2
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => {{1,3,6,7},{2},{4,5}} => 1
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => {{1,2,7},{3},{4,5,6}} => 2
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => {{1,7},{2},{3,5,6},{4}} => 2
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => {{1,4,5,6,7},{2},{3}} => 2
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [7,3,4,1,2,5,6] => {{1,7},{2,3,4},{5},{6}} => 2
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [2,7,5,1,3,4,6] => {{1,2,7},{3,5},{4},{6}} => 2
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [7,3,1,5,2,4,6] => {{1,7},{2,3},{4,5},{6}} => 1
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [5,4,1,2,7,3,6] => {{1,5,7},{2,4},{3},{6}} => 2
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => {{1,7},{2},{3,4,5},{6}} => 2
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => {{1,2,6,7},{3,4},{5}} => 3
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [4,3,1,6,2,7,5] => {{1,4,6,7},{2,3},{5}} => 2
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [2,3,7,1,6,4,5] => {{1,2,3,7},{4},{5,6}} => 2
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}} => 3
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => {{1,3,7},{2},{4,6},{5}} => 2
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => {{1,2,4,7},{3},{5,6}} => 3
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => {{1,7},{2},{3,4},{5,6}} => 3
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [5,3,1,2,6,7,4] => {{1,5,6,7},{2,3},{4}} => 1
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => {{1,2,6,7},{3},{4,5}} => 2
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => {{1,6,7},{2},{3,5},{4}} => 2
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => {{1,7},{2},{3},{4,5,6}} => 2
[5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => {{1,2,7},{3,4},{5},{6}} => 3
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => [4,3,1,7,2,5,6] => {{1,4,7},{2,3},{5},{6}} => 2
[5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => [2,3,5,1,7,4,6] => {{1,2,3,5,7},{4},{6}} => 1
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [5,7,1,2,3,4,6] => {{1,5},{2,7},{3},{4},{6}} => 2
[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => {{1,3,7},{2},{4,5},{6}} => 1
[5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => {{1,2,4,5,7},{3},{6}} => 2
[5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => {{1,5,7},{2},{3,4},{6}} => 2
[4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,6,1,4,7,5] => {{1,2,3,6,7},{4},{5}} => 2
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [6,4,1,2,3,7,5] => {{1,6,7},{2,4},{3},{5}} => 3
[4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => {{1,3,4,6,7},{2},{5}} => 2
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [7,3,1,2,6,4,5] => {{1,7},{2,3},{4},{5,6}} => 2
[4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => [2,7,1,6,3,4,5] => {{1,2,7},{3},{4,6},{5}} => 3
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [6,1,7,2,3,4,5] => {{1,6},{2},{3,7},{4},{5}} => 3
[4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => {{1,4,7},{2},{3},{5,6}} => 3
[3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => {{1,2,5,6,7},{3},{4}} => 2
[3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => {{1,3,5,6,7},{2},{4}} => 1
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => {{1,6,7},{2},{3},{4,5}} => 2
>>> Load all 132 entries. <<<
[5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => {{1,2,3,7},{4},{5},{6}} => 2
[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => {{1,7},{2,4},{3},{5},{6}} => 3
[5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [3,1,4,7,2,5,6] => {{1,3,4,7},{2},{5},{6}} => 2
[5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,1,2,7,4,6] => {{1,5,7},{2,3},{4},{6}} => 1
[5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0] => [2,7,1,5,3,4,6] => {{1,2,7},{3},{4,5},{6}} => 2
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => {{1,7},{2},{3,5},{4},{6}} => 2
[5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [4,1,2,5,7,3,6] => {{1,4,5,7},{2},{3},{6}} => 2
[4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => [6,3,1,2,4,7,5] => {{1,6,7},{2,3},{4},{5}} => 2
[4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => {{1,2,4,6,7},{3},{5}} => 3
[4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => {{1,6,7},{2},{3,4},{5}} => 3
[4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => {{1,2,7},{3},{4},{5,6}} => 3
[4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => {{1,3,7},{2},{4},{5,6}} => 2
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => {{1,7},{2},{3},{4,6},{5}} => 3
[3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => {{1,5,6,7},{2},{3},{4}} => 2
[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => {{1,7},{2,3},{4},{5},{6}} => 2
[5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => {{1,2,4,7},{3},{5},{6}} => 3
[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => {{1,7},{2},{3,4},{5},{6}} => 3
[5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,5,1,3,7,4,6] => {{1,2,5,7},{3},{4},{6}} => 2
[5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => {{1,3,5,7},{2},{4},{6}} => 1
[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => {{1,7},{2},{3},{4,5},{6}} => 2
[4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,6,1,3,4,7,5] => {{1,2,6,7},{3},{4},{5}} => 3
[4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => {{1,3,6,7},{2},{4},{5}} => 2
[4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => {{1,4,6,7},{2},{3},{5}} => 3
[4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => {{1,7},{2},{3},{4},{5,6}} => 3
[5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => {{1,2,7},{3},{4},{5},{6}} => 3
[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [3,1,7,2,4,5,6] => {{1,3,7},{2},{4},{5},{6}} => 2
[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [4,1,2,7,3,5,6] => {{1,4,7},{2},{3},{5},{6}} => 3
[5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [5,1,2,3,7,4,6] => {{1,5,7},{2},{3},{4},{6}} => 2
[4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => {{1,6,7},{2},{3},{4},{5}} => 3
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => {{1,7},{2},{3},{4},{5},{6}} => 3
[] => [] => [1] => {{1}} => 1
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
click to show known generating functions       
Description
The number of blocks with odd minimum.
See St000746The number of pairs with odd minimum in a perfect matching. for the analogous statistic on perfect matchings.
Map
weak exceedance partition
Description
The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.