Identifier
Values
[1,1] => [1] => 1
[2,1] => [1] => 1
[1,1,1] => [1,1] => 1
[3,1] => [1] => 1
[2,2] => [2] => 1
[2,1,1] => [1,1] => 1
[1,1,1,1] => [1,1,1] => 1
[4,1] => [1] => 1
[3,2] => [2] => 1
[3,1,1] => [1,1] => 1
[2,2,1] => [2,1] => 1
[2,1,1,1] => [1,1,1] => 1
[1,1,1,1,1] => [1,1,1,1] => 1
[5,1] => [1] => 1
[4,2] => [2] => 1
[4,1,1] => [1,1] => 1
[3,3] => [3] => 2
[3,2,1] => [2,1] => 1
[3,1,1,1] => [1,1,1] => 1
[2,2,2] => [2,2] => 1
[2,2,1,1] => [2,1,1] => 1
[2,1,1,1,1] => [1,1,1,1] => 1
[1,1,1,1,1,1] => [1,1,1,1,1] => 1
[6,1] => [1] => 1
[5,2] => [2] => 1
[5,1,1] => [1,1] => 1
[4,3] => [3] => 2
[4,2,1] => [2,1] => 1
[4,1,1,1] => [1,1,1] => 1
[3,3,1] => [3,1] => 2
[3,2,2] => [2,2] => 1
[3,2,1,1] => [2,1,1] => 1
[3,1,1,1,1] => [1,1,1,1] => 1
[2,2,2,1] => [2,2,1] => 1
[2,2,1,1,1] => [2,1,1,1] => 1
[2,1,1,1,1,1] => [1,1,1,1,1] => 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[7,1] => [1] => 1
[6,2] => [2] => 1
[6,1,1] => [1,1] => 1
[5,3] => [3] => 2
[5,2,1] => [2,1] => 1
[5,1,1,1] => [1,1,1] => 1
[4,4] => [4] => 5
[4,3,1] => [3,1] => 2
[4,2,2] => [2,2] => 1
[4,2,1,1] => [2,1,1] => 1
[4,1,1,1,1] => [1,1,1,1] => 1
[3,3,2] => [3,2] => 2
[3,3,1,1] => [3,1,1] => 2
[3,2,2,1] => [2,2,1] => 1
[3,2,1,1,1] => [2,1,1,1] => 1
[3,1,1,1,1,1] => [1,1,1,1,1] => 1
[2,2,2,2] => [2,2,2] => 1
[2,2,2,1,1] => [2,2,1,1] => 1
[2,2,1,1,1,1] => [2,1,1,1,1] => 1
[2,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[8,1] => [1] => 1
[7,2] => [2] => 1
[7,1,1] => [1,1] => 1
[6,3] => [3] => 2
[6,2,1] => [2,1] => 1
[6,1,1,1] => [1,1,1] => 1
[5,4] => [4] => 5
[5,3,1] => [3,1] => 2
[5,2,2] => [2,2] => 1
[5,2,1,1] => [2,1,1] => 1
[5,1,1,1,1] => [1,1,1,1] => 1
[4,4,1] => [4,1] => 5
[4,3,2] => [3,2] => 2
[4,3,1,1] => [3,1,1] => 2
[4,2,2,1] => [2,2,1] => 1
[4,2,1,1,1] => [2,1,1,1] => 1
[4,1,1,1,1,1] => [1,1,1,1,1] => 1
[3,3,3] => [3,3] => 4
[3,3,2,1] => [3,2,1] => 2
[3,3,1,1,1] => [3,1,1,1] => 2
[3,2,2,2] => [2,2,2] => 1
[3,2,2,1,1] => [2,2,1,1] => 1
[3,2,1,1,1,1] => [2,1,1,1,1] => 1
[3,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[2,2,2,2,1] => [2,2,2,1] => 1
[2,2,2,1,1,1] => [2,2,1,1,1] => 1
[2,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[9,1] => [1] => 1
[8,2] => [2] => 1
[8,1,1] => [1,1] => 1
[7,3] => [3] => 2
[7,2,1] => [2,1] => 1
[7,1,1,1] => [1,1,1] => 1
[6,4] => [4] => 5
[6,3,1] => [3,1] => 2
[6,2,2] => [2,2] => 1
[6,2,1,1] => [2,1,1] => 1
[6,1,1,1,1] => [1,1,1,1] => 1
[5,5] => [5] => 14
[5,4,1] => [4,1] => 5
[5,3,2] => [3,2] => 2
[5,3,1,1] => [3,1,1] => 2
>>> Load all 421 entries. <<<
[5,2,2,1] => [2,2,1] => 1
[5,2,1,1,1] => [2,1,1,1] => 1
[5,1,1,1,1,1] => [1,1,1,1,1] => 1
[4,4,2] => [4,2] => 5
[4,4,1,1] => [4,1,1] => 5
[4,3,3] => [3,3] => 4
[4,3,2,1] => [3,2,1] => 2
[4,3,1,1,1] => [3,1,1,1] => 2
[4,2,2,2] => [2,2,2] => 1
[4,2,2,1,1] => [2,2,1,1] => 1
[4,2,1,1,1,1] => [2,1,1,1,1] => 1
[4,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[3,3,3,1] => [3,3,1] => 4
[3,3,2,2] => [3,2,2] => 2
[3,3,2,1,1] => [3,2,1,1] => 2
[3,3,1,1,1,1] => [3,1,1,1,1] => 2
[3,2,2,2,1] => [2,2,2,1] => 1
[3,2,2,1,1,1] => [2,2,1,1,1] => 1
[3,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[10,1] => [1] => 1
[9,2] => [2] => 1
[9,1,1] => [1,1] => 1
[8,3] => [3] => 2
[8,2,1] => [2,1] => 1
[8,1,1,1] => [1,1,1] => 1
[7,4] => [4] => 5
[7,3,1] => [3,1] => 2
[7,2,2] => [2,2] => 1
[7,2,1,1] => [2,1,1] => 1
[7,1,1,1,1] => [1,1,1,1] => 1
[6,5] => [5] => 14
[6,4,1] => [4,1] => 5
[6,3,2] => [3,2] => 2
[6,3,1,1] => [3,1,1] => 2
[6,2,2,1] => [2,2,1] => 1
[6,2,1,1,1] => [2,1,1,1] => 1
[6,1,1,1,1,1] => [1,1,1,1,1] => 1
[5,5,1] => [5,1] => 14
[5,4,2] => [4,2] => 5
[5,4,1,1] => [4,1,1] => 5
[5,3,3] => [3,3] => 4
[5,3,2,1] => [3,2,1] => 2
[5,3,1,1,1] => [3,1,1,1] => 2
[5,2,2,2] => [2,2,2] => 1
[5,2,2,1,1] => [2,2,1,1] => 1
[5,2,1,1,1,1] => [2,1,1,1,1] => 1
[5,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[4,4,3] => [4,3] => 10
[4,4,2,1] => [4,2,1] => 5
[4,4,1,1,1] => [4,1,1,1] => 5
[4,3,3,1] => [3,3,1] => 4
[4,3,2,2] => [3,2,2] => 2
[4,3,2,1,1] => [3,2,1,1] => 2
[4,3,1,1,1,1] => [3,1,1,1,1] => 2
[4,2,2,2,1] => [2,2,2,1] => 1
[4,2,2,1,1,1] => [2,2,1,1,1] => 1
[4,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[4,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[11,1] => [1] => 1
[10,2] => [2] => 1
[10,1,1] => [1,1] => 1
[9,3] => [3] => 2
[9,2,1] => [2,1] => 1
[9,1,1,1] => [1,1,1] => 1
[8,4] => [4] => 5
[8,3,1] => [3,1] => 2
[8,2,2] => [2,2] => 1
[8,2,1,1] => [2,1,1] => 1
[8,1,1,1,1] => [1,1,1,1] => 1
[7,5] => [5] => 14
[7,4,1] => [4,1] => 5
[7,3,2] => [3,2] => 2
[7,3,1,1] => [3,1,1] => 2
[7,2,2,1] => [2,2,1] => 1
[7,2,1,1,1] => [2,1,1,1] => 1
[7,1,1,1,1,1] => [1,1,1,1,1] => 1
[6,6] => [6] => 42
[6,5,1] => [5,1] => 14
[6,4,2] => [4,2] => 5
[6,4,1,1] => [4,1,1] => 5
[6,3,3] => [3,3] => 4
[6,3,2,1] => [3,2,1] => 2
[6,3,1,1,1] => [3,1,1,1] => 2
[6,2,2,2] => [2,2,2] => 1
[6,2,2,1,1] => [2,2,1,1] => 1
[6,2,1,1,1,1] => [2,1,1,1,1] => 1
[6,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[5,5,2] => [5,2] => 14
[5,5,1,1] => [5,1,1] => 14
[5,4,3] => [4,3] => 10
[5,4,2,1] => [4,2,1] => 5
[5,4,1,1,1] => [4,1,1,1] => 5
[5,3,3,1] => [3,3,1] => 4
[5,3,2,2] => [3,2,2] => 2
[5,3,2,1,1] => [3,2,1,1] => 2
[5,3,1,1,1,1] => [3,1,1,1,1] => 2
[5,2,2,2,1] => [2,2,2,1] => 1
[5,2,2,1,1,1] => [2,2,1,1,1] => 1
[5,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[5,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[12,1] => [1] => 1
[11,2] => [2] => 1
[11,1,1] => [1,1] => 1
[10,3] => [3] => 2
[10,2,1] => [2,1] => 1
[10,1,1,1] => [1,1,1] => 1
[9,4] => [4] => 5
[9,3,1] => [3,1] => 2
[9,2,2] => [2,2] => 1
[9,2,1,1] => [2,1,1] => 1
[9,1,1,1,1] => [1,1,1,1] => 1
[8,5] => [5] => 14
[8,4,1] => [4,1] => 5
[8,3,2] => [3,2] => 2
[8,3,1,1] => [3,1,1] => 2
[8,2,2,1] => [2,2,1] => 1
[8,2,1,1,1] => [2,1,1,1] => 1
[8,1,1,1,1,1] => [1,1,1,1,1] => 1
[7,6] => [6] => 42
[7,5,1] => [5,1] => 14
[7,4,2] => [4,2] => 5
[7,4,1,1] => [4,1,1] => 5
[7,3,3] => [3,3] => 4
[7,3,2,1] => [3,2,1] => 2
[7,3,1,1,1] => [3,1,1,1] => 2
[7,2,2,2] => [2,2,2] => 1
[7,2,2,1,1] => [2,2,1,1] => 1
[7,2,1,1,1,1] => [2,1,1,1,1] => 1
[7,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[6,6,1] => [6,1] => 42
[6,5,2] => [5,2] => 14
[6,5,1,1] => [5,1,1] => 14
[6,4,3] => [4,3] => 10
[6,4,2,1] => [4,2,1] => 5
[6,4,1,1,1] => [4,1,1,1] => 5
[6,3,3,1] => [3,3,1] => 4
[6,3,2,2] => [3,2,2] => 2
[6,3,2,1,1] => [3,2,1,1] => 2
[6,3,1,1,1,1] => [3,1,1,1,1] => 2
[6,2,2,2,1] => [2,2,2,1] => 1
[6,2,2,1,1,1] => [2,2,1,1,1] => 1
[6,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[6,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[13,1] => [1] => 1
[12,2] => [2] => 1
[12,1,1] => [1,1] => 1
[11,3] => [3] => 2
[11,2,1] => [2,1] => 1
[11,1,1,1] => [1,1,1] => 1
[10,4] => [4] => 5
[10,3,1] => [3,1] => 2
[10,2,2] => [2,2] => 1
[10,2,1,1] => [2,1,1] => 1
[10,1,1,1,1] => [1,1,1,1] => 1
[9,5] => [5] => 14
[9,4,1] => [4,1] => 5
[9,3,2] => [3,2] => 2
[9,3,1,1] => [3,1,1] => 2
[9,2,2,1] => [2,2,1] => 1
[9,2,1,1,1] => [2,1,1,1] => 1
[9,1,1,1,1,1] => [1,1,1,1,1] => 1
[8,6] => [6] => 42
[8,5,1] => [5,1] => 14
[8,4,2] => [4,2] => 5
[8,4,1,1] => [4,1,1] => 5
[8,3,3] => [3,3] => 4
[8,3,2,1] => [3,2,1] => 2
[8,3,1,1,1] => [3,1,1,1] => 2
[8,2,2,2] => [2,2,2] => 1
[8,2,2,1,1] => [2,2,1,1] => 1
[8,2,1,1,1,1] => [2,1,1,1,1] => 1
[8,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[7,7] => [7] => 132
[7,6,1] => [6,1] => 42
[7,5,2] => [5,2] => 14
[7,5,1,1] => [5,1,1] => 14
[7,4,3] => [4,3] => 10
[7,4,2,1] => [4,2,1] => 5
[7,4,1,1,1] => [4,1,1,1] => 5
[7,3,3,1] => [3,3,1] => 4
[7,3,2,2] => [3,2,2] => 2
[7,3,2,1,1] => [3,2,1,1] => 2
[7,3,1,1,1,1] => [3,1,1,1,1] => 2
[7,2,2,2,1] => [2,2,2,1] => 1
[7,2,2,1,1,1] => [2,2,1,1,1] => 1
[7,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[7,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[14,1] => [1] => 1
[13,2] => [2] => 1
[13,1,1] => [1,1] => 1
[12,3] => [3] => 2
[12,2,1] => [2,1] => 1
[12,1,1,1] => [1,1,1] => 1
[11,4] => [4] => 5
[11,3,1] => [3,1] => 2
[11,2,2] => [2,2] => 1
[11,2,1,1] => [2,1,1] => 1
[11,1,1,1,1] => [1,1,1,1] => 1
[10,5] => [5] => 14
[10,4,1] => [4,1] => 5
[10,3,2] => [3,2] => 2
[10,3,1,1] => [3,1,1] => 2
[10,2,2,1] => [2,2,1] => 1
[10,2,1,1,1] => [2,1,1,1] => 1
[10,1,1,1,1,1] => [1,1,1,1,1] => 1
[9,6] => [6] => 42
[9,5,1] => [5,1] => 14
[9,4,2] => [4,2] => 5
[9,4,1,1] => [4,1,1] => 5
[9,3,3] => [3,3] => 4
[9,3,2,1] => [3,2,1] => 2
[9,3,1,1,1] => [3,1,1,1] => 2
[9,2,2,2] => [2,2,2] => 1
[9,2,2,1,1] => [2,2,1,1] => 1
[9,2,1,1,1,1] => [2,1,1,1,1] => 1
[9,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[8,7] => [7] => 132
[8,6,1] => [6,1] => 42
[8,5,2] => [5,2] => 14
[8,5,1,1] => [5,1,1] => 14
[8,4,3] => [4,3] => 10
[8,4,2,1] => [4,2,1] => 5
[8,4,1,1,1] => [4,1,1,1] => 5
[8,3,3,1] => [3,3,1] => 4
[8,3,2,2] => [3,2,2] => 2
[8,3,2,1,1] => [3,2,1,1] => 2
[8,3,1,1,1,1] => [3,1,1,1,1] => 2
[8,2,2,2,1] => [2,2,2,1] => 1
[8,2,2,1,1,1] => [2,2,1,1,1] => 1
[8,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[8,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[15,1] => [1] => 1
[14,2] => [2] => 1
[14,1,1] => [1,1] => 1
[13,3] => [3] => 2
[13,2,1] => [2,1] => 1
[13,1,1,1] => [1,1,1] => 1
[12,4] => [4] => 5
[12,3,1] => [3,1] => 2
[12,2,2] => [2,2] => 1
[12,2,1,1] => [2,1,1] => 1
[12,1,1,1,1] => [1,1,1,1] => 1
[11,5] => [5] => 14
[11,4,1] => [4,1] => 5
[11,3,2] => [3,2] => 2
[11,3,1,1] => [3,1,1] => 2
[11,2,2,1] => [2,2,1] => 1
[11,2,1,1,1] => [2,1,1,1] => 1
[11,1,1,1,1,1] => [1,1,1,1,1] => 1
[10,6] => [6] => 42
[10,5,1] => [5,1] => 14
[10,4,2] => [4,2] => 5
[10,4,1,1] => [4,1,1] => 5
[10,3,3] => [3,3] => 4
[10,3,2,1] => [3,2,1] => 2
[10,3,1,1,1] => [3,1,1,1] => 2
[10,2,2,2] => [2,2,2] => 1
[10,2,2,1,1] => [2,2,1,1] => 1
[10,2,1,1,1,1] => [2,1,1,1,1] => 1
[10,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[9,7] => [7] => 132
[9,6,1] => [6,1] => 42
[9,5,2] => [5,2] => 14
[9,5,1,1] => [5,1,1] => 14
[9,4,3] => [4,3] => 10
[9,4,2,1] => [4,2,1] => 5
[9,4,1,1,1] => [4,1,1,1] => 5
[9,3,3,1] => [3,3,1] => 4
[9,3,2,2] => [3,2,2] => 2
[9,3,2,1,1] => [3,2,1,1] => 2
[9,3,1,1,1,1] => [3,1,1,1,1] => 2
[9,2,2,2,1] => [2,2,2,1] => 1
[9,2,2,1,1,1] => [2,2,1,1,1] => 1
[9,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[9,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
[16,1] => [1] => 1
[15,2] => [2] => 1
[15,1,1] => [1,1] => 1
[14,3] => [3] => 2
[14,2,1] => [2,1] => 1
[14,1,1,1] => [1,1,1] => 1
[13,4] => [4] => 5
[13,3,1] => [3,1] => 2
[13,2,2] => [2,2] => 1
[13,2,1,1] => [2,1,1] => 1
[13,1,1,1,1] => [1,1,1,1] => 1
[12,5] => [5] => 14
[12,4,1] => [4,1] => 5
[12,3,2] => [3,2] => 2
[12,3,1,1] => [3,1,1] => 2
[12,2,2,1] => [2,2,1] => 1
[12,2,1,1,1] => [2,1,1,1] => 1
[12,1,1,1,1,1] => [1,1,1,1,1] => 1
[11,6] => [6] => 42
[11,5,1] => [5,1] => 14
[11,4,2] => [4,2] => 5
[11,4,1,1] => [4,1,1] => 5
[11,3,3] => [3,3] => 4
[11,3,2,1] => [3,2,1] => 2
[11,3,1,1,1] => [3,1,1,1] => 2
[11,2,2,2] => [2,2,2] => 1
[11,2,2,1,1] => [2,2,1,1] => 1
[11,2,1,1,1,1] => [2,1,1,1,1] => 1
[11,1,1,1,1,1,1] => [1,1,1,1,1,1] => 1
[10,7] => [7] => 132
[10,6,1] => [6,1] => 42
[10,5,2] => [5,2] => 14
[10,5,1,1] => [5,1,1] => 14
[10,4,3] => [4,3] => 10
[10,4,2,1] => [4,2,1] => 5
[10,4,1,1,1] => [4,1,1,1] => 5
[10,3,3,1] => [3,3,1] => 4
[10,3,2,2] => [3,2,2] => 2
[10,3,2,1,1] => [3,2,1,1] => 2
[10,3,1,1,1,1] => [3,1,1,1,1] => 2
[10,2,2,2,1] => [2,2,2,1] => 1
[10,2,2,1,1,1] => [2,2,1,1,1] => 1
[10,2,1,1,1,1,1] => [2,1,1,1,1,1] => 1
[10,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 1
search for individual values
searching the database for the individual values of this statistic
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$ (a_1, b_1),\dots,(a_r, b_r) $$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Map
first row removal
Description
Removes the first entry of an integer partition