Identifier
Values
[1] => [1,0,1,0] => [2,1] => 1
[2] => [1,1,0,0,1,0] => [3,1,2] => 1
[1,1] => [1,0,1,1,0,0] => [2,3,1] => 1
[3] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
[2,1] => [1,0,1,0,1,0] => [3,2,1] => 2
[1,1,1] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[4] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[3,1] => [1,1,0,1,0,0,1,0] => [4,2,1,3] => 3
[2,2] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [3,2,4,1] => 3
[1,1,1,1] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[5] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 1
[4,1] => [1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 4
[3,2] => [1,1,0,0,1,0,1,0] => [4,3,1,2] => 5
[3,1,1] => [1,0,1,1,0,0,1,0] => [4,2,3,1] => 6
[2,2,1] => [1,0,1,0,1,1,0,0] => [3,4,2,1] => 5
[2,1,1,1] => [1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 4
[1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [7,1,2,3,4,5,6] => 1
[5,1] => [1,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,3,4,5] => 5
[4,2] => [1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 9
[4,1,1] => [1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 10
[3,3] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 5
[3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => 16
[3,1,1,1] => [1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 10
[2,2,2] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 5
[2,2,1,1] => [1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 9
[2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,4,5,6,1] => 5
[1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
[6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [7,2,1,3,4,5,6] => 6
[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,2,4,5] => 14
[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => 15
[4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 14
[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 35
[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 20
[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 21
[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 21
[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 35
[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => 15
[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 14
[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => 14
[2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => 6
[6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [7,3,1,2,4,5,6] => 20
[6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [7,2,3,1,4,5,6] => 21
[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [6,4,1,2,3,5] => 28
[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1,4,5] => 64
[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => 35
[4,4] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 14
[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => 70
[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 56
[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 90
[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => 35
[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 42
[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 56
[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 70
[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => 64
[3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [4,2,3,5,6,7,1] => 21
[2,2,2,2] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => 14
[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => 28
[2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [3,4,2,5,6,7,1] => 20
[6,3] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [7,4,1,2,3,5,6] => 48
[6,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [7,3,2,1,4,5,6] => 105
[6,1,1,1] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [7,2,3,4,1,5,6] => 56
[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 42
[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [6,4,2,1,3,5] => 162
[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [6,3,4,1,2,5] => 120
[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [6,3,2,4,1,5] => 189
[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => 70
[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [5,6,2,1,3,4] => 84
[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 168
[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 216
[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 216
[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => 189
[4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [5,2,3,4,6,7,1] => 56
[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => 42
[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 168
[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => 120
[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [4,3,5,6,1,2] => 84
[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => 162
[3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [4,3,2,5,6,7,1] => 105
[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => 42
[2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [3,4,5,2,6,7,1] => 48
[6,4] => [1,1,1,1,1,0,0,0,0,1,0,0,1,0] => [7,5,1,2,3,4,6] => 90
[6,3,1] => [1,1,1,1,0,1,0,0,1,0,0,0,1,0] => [7,4,2,1,3,5,6] => 315
[6,2,2] => [1,1,1,1,0,0,1,1,0,0,0,0,1,0] => [7,3,4,1,2,5,6] => 225
[6,2,1,1] => [1,1,1,0,1,1,0,1,0,0,0,0,1,0] => [7,3,2,4,1,5,6] => 350
[6,1,1,1,1] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0] => [7,2,3,4,5,1,6] => 126
[5,5] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [6,7,1,2,3,4,5] => 42
[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [6,5,2,1,3,4] => 288
[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [6,4,3,1,2,5] => 450
[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [6,4,2,3,1,5] => 567
[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [6,3,4,2,1,5] => 525
[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => 448
[5,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [6,2,3,4,5,7,1] => 126
[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [5,6,3,1,2,4] => 252
[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [5,6,2,3,1,4] => 300
[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [5,4,6,1,2,3] => 210
[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 768
[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => 525
[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [5,3,4,6,1,2] => 300
[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => 567
>>> Load all 170 entries. <<<
[4,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,0,1,0,0,0] => [5,3,2,4,6,7,1] => 350
[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [4,5,6,2,1,3] => 210
[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => [4,5,3,6,1,2] => 252
[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => 450
[3,3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0] => [4,5,2,3,6,7,1] => 225
[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => 288
[3,2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [4,3,5,2,6,7,1] => 315
[2,2,2,2,2] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [3,4,5,6,7,1,2] => 42
[2,2,2,2,1,1] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [3,4,5,6,2,7,1] => 90
[6,5] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => [7,6,1,2,3,4,5] => 132
[5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [6,5,3,1,2,4] => 990
[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => [6,5,2,3,1,4] => 1155
[5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => 660
[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [6,4,3,2,1,5] => 2310
[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => 1540
[5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => 825
[5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => 1540
[4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => 462
[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [5,6,3,2,1,4] => 1320
[4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => 825
[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [5,4,6,2,1,3] => 1188
[4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => [5,4,3,6,1,2] => 1320
[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => 2310
[4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => 1155
[3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => 462
[3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => 660
[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => 990
[5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => 2112
[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [6,5,3,2,1,4] => 5775
[5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => 3520
[5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0] => [6,4,5,2,1,3] => 4158
[5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0] => [6,4,3,5,1,2] => 4455
[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => 7700
[5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => 3520
[4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [5,6,7,1,2,3,4] => 462
[4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => [5,6,4,2,1,3] => 2970
[4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => 2640
[4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => 4455
[4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => [5,4,6,3,1,2] => 2970
[4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => 4158
[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => 5775
[3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => 2112
[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [6,5,4,2,1,3] => 15015
[5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => 12870
[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => 21450
[5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => 11583
[5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => 16016
[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => 21450
[4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => 8580
[4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => 11583
[4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => 12870
[4,3,3,3] => [1,1,1,0,0,0,1,1,1,0,1,0,0,0] => [5,4,6,7,1,2,3] => 3432
[4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => 15015
[6,4,4] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [7,5,6,1,2,3,4] => 9009
[5,5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,1,0,0] => [6,7,3,4,1,2,5] => 21450
[5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => 48048
[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => 64064
[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => 68640
[5,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,1,0,0] => [6,4,5,7,1,2,3] => 15015
[5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => 64064
[4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => 48048
[6,5,4] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [7,6,5,1,2,3,4] => 30030
[6,3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [7,4,5,6,1,2,3] => 50050
[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => 292864
[5,5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [6,7,4,5,1,2,3] => 171600
[6,5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [7,6,4,5,1,2,3] => 972400
[5,4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [6,5,7,3,4,1,2] => 1361360
[] => [] => [] => 1
[6,4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [7,5,6,3,4,1,2] => 6534528
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Description
The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation $[3,2,1]$, which are $(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].