Identifier
Values
[1] => [1] => 0
[2] => [2] => 1
[1,1] => [1,1] => 0
[3] => [2,1] => 1
[2,1] => [3] => 0
[1,1,1] => [1,1,1] => 0
[4] => [2,2] => 2
[3,1] => [2,1,1] => 1
[2,2] => [4] => 1
[2,1,1] => [3,1] => 0
[1,1,1,1] => [1,1,1,1] => 0
[5] => [2,2,1] => 2
[4,1] => [3,2] => 1
[3,2] => [4,1] => 1
[3,1,1] => [2,1,1,1] => 1
[2,2,1] => [5] => 0
[2,1,1,1] => [3,1,1] => 0
[1,1,1,1,1] => [1,1,1,1,1] => 0
[6] => [2,2,2] => 3
[5,1] => [2,2,1,1] => 2
[4,2] => [4,2] => 2
[4,1,1] => [4,1,1] => 1
[3,3] => [3,2,1] => 1
[3,2,1] => [3,3] => 0
[3,1,1,1] => [2,1,1,1,1] => 1
[2,2,2] => [6] => 1
[2,2,1,1] => [5,1] => 0
[2,1,1,1,1] => [3,1,1,1] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[7] => [2,2,2,1] => 3
[6,1] => [3,2,2] => 2
[5,2] => [4,2,1] => 2
[5,1,1] => [2,2,1,1,1] => 2
[4,3] => [4,3] => 1
[4,2,1] => [5,2] => 1
[4,1,1,1] => [4,1,1,1] => 1
[3,3,1] => [3,2,1,1] => 1
[3,2,2] => [6,1] => 1
[3,2,1,1] => [3,3,1] => 0
[3,1,1,1,1] => [2,1,1,1,1,1] => 1
[2,2,2,1] => [7] => 0
[2,2,1,1,1] => [5,1,1] => 0
[2,1,1,1,1,1] => [3,1,1,1,1] => 0
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 0
[8] => [2,2,2,2] => 4
[7,1] => [2,2,2,1,1] => 3
[6,2] => [4,2,2] => 3
[6,1,1] => [4,2,1,1] => 2
[5,3] => [3,2,2,1] => 2
[5,2,1] => [3,3,2] => 1
[5,1,1,1] => [2,2,1,1,1,1] => 2
[4,4] => [4,4] => 2
[4,3,1] => [4,3,1] => 1
[4,2,2] => [6,2] => 2
[4,2,1,1] => [6,1,1] => 1
[4,1,1,1,1] => [4,1,1,1,1] => 1
[3,3,2] => [5,2,1] => 1
[3,3,1,1] => [3,2,1,1,1] => 1
[3,2,2,1] => [5,3] => 0
[3,2,1,1,1] => [3,3,1,1] => 0
[3,1,1,1,1,1] => [2,1,1,1,1,1,1] => 1
[2,2,2,2] => [8] => 1
[2,2,2,1,1] => [7,1] => 0
[2,2,1,1,1,1] => [5,1,1,1] => 0
[2,1,1,1,1,1,1] => [3,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1] => 0
[9] => [2,2,2,2,1] => 4
[8,1] => [3,2,2,2] => 3
[7,2] => [4,2,2,1] => 3
[7,1,1] => [2,2,2,1,1,1] => 3
[6,3] => [4,3,2] => 2
[6,2,1] => [5,2,2] => 2
[6,1,1,1] => [4,2,1,1,1] => 2
[5,4] => [4,4,1] => 2
[5,3,1] => [3,2,2,1,1] => 2
[5,2,2] => [6,2,1] => 2
[5,2,1,1] => [4,3,1,1] => 1
[5,1,1,1,1] => [2,2,1,1,1,1,1] => 2
[4,4,1] => [5,4] => 1
[4,3,2] => [6,3] => 1
[4,3,1,1] => [5,2,1,1] => 1
[4,2,2,1] => [7,2] => 1
[4,2,1,1,1] => [6,1,1,1] => 1
[4,1,1,1,1,1] => [4,1,1,1,1,1] => 1
[3,3,3] => [3,3,2,1] => 1
[3,3,2,1] => [3,3,3] => 0
[3,3,1,1,1] => [3,2,1,1,1,1] => 1
[3,2,2,2] => [8,1] => 1
[3,2,2,1,1] => [5,3,1] => 0
[3,2,1,1,1,1] => [3,3,1,1,1] => 0
[3,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1] => 1
[2,2,2,2,1] => [9] => 0
[2,2,2,1,1,1] => [7,1,1] => 0
[2,2,1,1,1,1,1] => [5,1,1,1,1] => 0
[2,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1] => 0
[10] => [2,2,2,2,2] => 5
[9,1] => [2,2,2,2,1,1] => 4
[8,2] => [4,2,2,2] => 4
[8,1,1] => [4,2,2,1,1] => 3
[7,3] => [3,2,2,2,1] => 3
>>> Load all 352 entries. <<<
[7,2,1] => [3,3,2,2] => 2
[7,1,1,1] => [2,2,2,1,1,1,1] => 3
[6,4] => [4,4,2] => 3
[6,3,1] => [4,4,1,1] => 2
[6,2,2] => [6,2,2] => 3
[6,2,1,1] => [6,2,1,1] => 2
[6,1,1,1,1] => [4,2,1,1,1,1] => 2
[5,5] => [4,3,2,1] => 2
[5,4,1] => [4,3,3] => 1
[5,3,2] => [5,2,2,1] => 2
[5,3,1,1] => [3,2,2,1,1,1] => 2
[5,2,2,1] => [5,3,2] => 1
[5,2,1,1,1] => [4,3,1,1,1] => 1
[5,1,1,1,1,1] => [2,2,1,1,1,1,1,1] => 2
[4,4,2] => [6,4] => 2
[4,4,1,1] => [6,3,1] => 1
[4,3,3] => [5,4,1] => 1
[4,3,2,1] => [5,5] => 0
[4,3,1,1,1] => [5,2,1,1,1] => 1
[4,2,2,2] => [8,2] => 2
[4,2,2,1,1] => [8,1,1] => 1
[4,2,1,1,1,1] => [6,1,1,1,1] => 1
[4,1,1,1,1,1,1] => [4,1,1,1,1,1,1] => 1
[3,3,3,1] => [3,3,2,1,1] => 1
[3,3,2,2] => [7,2,1] => 1
[3,3,2,1,1] => [3,3,3,1] => 0
[3,3,1,1,1,1] => [3,2,1,1,1,1,1] => 1
[3,2,2,2,1] => [7,3] => 0
[3,2,2,1,1,1] => [5,3,1,1] => 0
[3,2,1,1,1,1,1] => [3,3,1,1,1,1] => 0
[3,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2] => [10] => 1
[2,2,2,2,1,1] => [9,1] => 0
[2,2,2,1,1,1,1] => [7,1,1,1] => 0
[2,2,1,1,1,1,1,1] => [5,1,1,1,1,1] => 0
[2,1,1,1,1,1,1,1,1] => [3,1,1,1,1,1,1,1] => 0
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => 0
[11] => [2,2,2,2,2,1] => 5
[10,1] => [3,2,2,2,2] => 4
[9,2] => [4,2,2,2,1] => 4
[8,3] => [4,3,2,2] => 3
[8,2,1] => [5,2,2,2] => 3
[7,4] => [4,4,2,1] => 3
[7,2,1,1] => [4,4,1,1,1] => 2
[7,1,1,1,1] => [2,2,2,1,1,1,1,1] => 3
[6,5] => [4,4,3] => 2
[6,4,1] => [5,4,2] => 2
[6,3,1,1] => [5,2,2,1,1] => 2
[6,2,2,1] => [7,2,2] => 2
[5,5,1] => [4,3,2,1,1] => 2
[5,4,2] => [6,4,1] => 2
[5,4,1,1] => [4,3,3,1] => 1
[5,3,3] => [3,3,2,2,1] => 2
[5,3,2,1] => [3,3,3,2] => 1
[4,4,3] => [6,5] => 1
[4,4,2,1] => [7,4] => 1
[4,3,3,1] => [5,4,1,1] => 1
[4,3,2,2] => [8,3] => 1
[4,3,2,1,1] => [5,5,1] => 0
[4,3,1,1,1,1] => [5,2,1,1,1,1] => 1
[4,2,1,1,1,1,1] => [6,1,1,1,1,1] => 1
[3,3,3,2] => [5,3,2,1] => 1
[3,3,2,2,1] => [5,3,3] => 0
[3,3,2,1,1,1] => [3,3,3,1,1] => 0
[3,2,2,2,2] => [10,1] => 1
[3,2,2,1,1,1,1] => [5,3,1,1,1] => 0
[3,2,1,1,1,1,1,1] => [3,3,1,1,1,1,1] => 0
[3,1,1,1,1,1,1,1,1] => [2,1,1,1,1,1,1,1,1,1] => 1
[2,2,2,2,2,1] => [11] => 0
[2,2,2,2,1,1,1] => [9,1,1] => 0
[1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1] => 0
[12] => [2,2,2,2,2,2] => 6
[9,3] => [3,2,2,2,2,1] => 4
[9,2,1] => [3,3,2,2,2] => 3
[8,4] => [4,4,2,2] => 4
[8,3,1] => [4,4,2,1,1] => 3
[8,2,2] => [6,2,2,2] => 4
[8,2,1,1] => [6,2,2,1,1] => 3
[7,5] => [4,3,2,2,1] => 3
[7,4,1] => [4,3,3,2] => 2
[7,3,2] => [5,2,2,2,1] => 3
[7,2,2,1] => [5,3,2,2] => 2
[6,6] => [4,4,4] => 3
[6,5,1] => [4,4,3,1] => 2
[6,4,2] => [6,4,2] => 3
[6,3,3] => [5,4,2,1] => 2
[6,3,2,1] => [5,5,2] => 1
[6,3,1,1,1] => [5,2,2,1,1,1] => 2
[6,2,2,2] => [8,2,2] => 3
[5,4,3] => [5,4,3] => 1
[5,4,2,1] => [6,3,3] => 1
[5,4,1,1,1] => [5,4,1,1,1] => 1
[5,3,3,1] => [3,3,2,2,1,1] => 2
[5,3,2,1,1] => [4,3,3,1,1] => 1
[5,1,1,1,1,1,1,1] => [2,2,1,1,1,1,1,1,1,1] => 2
[4,4,4] => [6,6] => 2
[4,3,3,2] => [7,4,1] => 1
[4,3,3,1,1] => [5,3,2,1,1] => 1
[4,3,2,2,1] => [7,5] => 0
[4,2,2,2,2] => [10,2] => 2
[4,2,2,2,1,1] => [10,1,1] => 1
[3,3,3,3] => [3,3,3,2,1] => 1
[3,3,3,2,1] => [3,3,3,3] => 0
[3,3,2,2,1,1] => [5,3,3,1] => 0
[3,2,2,2,2,1] => [9,3] => 0
[2,2,2,2,2,2] => [12] => 1
[2,2,2,2,2,1,1] => [11,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1] => 0
[13] => [2,2,2,2,2,2,1] => 6
[12,1] => [3,2,2,2,2,2] => 5
[9,4] => [4,4,2,2,1] => 4
[8,5] => [4,4,3,2] => 3
[8,4,1] => [5,4,2,2] => 3
[7,6] => [4,4,4,1] => 3
[7,4,2] => [6,4,2,1] => 3
[7,4,1,1] => [4,4,3,1,1] => 2
[7,3,3] => [3,3,2,2,2,1] => 3
[7,3,2,1] => [3,3,3,2,2] => 2
[6,6,1] => [5,4,4] => 2
[6,4,2,1] => [7,4,2] => 2
[6,3,3,1] => [5,4,2,1,1] => 2
[6,3,2,2] => [8,3,2] => 2
[6,2,2,2,1] => [9,2,2] => 2
[5,5,3] => [4,3,3,2,1] => 2
[5,5,2,1] => [4,3,3,3] => 1
[5,4,4] => [6,6,1] => 2
[5,4,3,1] => [5,4,3,1] => 1
[5,4,1,1,1,1] => [5,4,1,1,1,1] => 1
[5,3,3,2] => [5,3,2,2,1] => 2
[5,3,2,2,1] => [5,3,3,2] => 1
[4,4,4,1] => [7,6] => 1
[4,4,3,2] => [8,5] => 1
[4,3,3,2,1] => [5,5,3] => 0
[4,3,3,1,1,1] => [5,3,2,1,1,1] => 1
[4,3,2,2,2] => [10,3] => 1
[4,3,2,2,1,1] => [7,5,1] => 0
[3,3,3,2,1,1] => [3,3,3,3,1] => 0
[3,3,3,1,1,1,1] => [3,3,2,1,1,1,1,1] => 1
[3,3,2,2,1,1,1] => [5,3,3,1,1] => 0
[3,2,2,2,2,2] => [12,1] => 1
[2,2,2,2,2,2,1] => [13] => 0
[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] => 0
[14] => [2,2,2,2,2,2,2] => 7
[11,2,1] => [3,3,2,2,2,2] => 4
[10,2,2] => [6,2,2,2,2] => 5
[9,5] => [4,3,2,2,2,1] => 4
[9,3,2] => [5,2,2,2,2,1] => 4
[9,2,2,1] => [5,3,2,2,2] => 3
[8,6] => [4,4,4,2] => 4
[8,4,2] => [6,4,2,2] => 4
[8,3,3] => [5,4,2,2,1] => 3
[7,7] => [4,4,3,2,1] => 3
[7,6,1] => [4,4,3,3] => 2
[7,4,3] => [5,4,3,2] => 2
[7,4,1,1,1] => [5,4,2,1,1,1] => 2
[6,6,2] => [6,4,4] => 3
[6,5,2,1] => [5,5,4] => 1
[6,4,4] => [6,6,2] => 3
[6,4,2,2] => [8,4,2] => 3
[6,3,2,2,1] => [7,5,2] => 1
[5,4,4,1] => [6,5,3] => 1
[5,4,3,2] => [7,4,3] => 1
[5,4,3,1,1] => [5,4,3,1,1] => 1
[5,3,3,3] => [3,3,3,2,2,1] => 2
[5,3,3,2,1] => [3,3,3,3,2] => 1
[4,4,4,2] => [8,6] => 2
[4,4,4,1,1] => [8,5,1] => 1
[4,4,3,2,1] => [7,7] => 0
[4,3,2,2,2,1] => [9,5] => 0
[4,3,2,1,1,1,1,1] => [5,5,1,1,1,1] => 0
[4,2,2,2,2,2] => [12,2] => 2
[3,3,3,3,2] => [5,3,3,2,1] => 1
[3,3,3,2,2,1] => [5,3,3,3] => 0
[3,3,3,2,1,1,1] => [3,3,3,3,1,1] => 0
[2,2,2,2,2,2,2] => [14] => 1
[2,2,2,2,2,2,1,1] => [13,1] => 0
[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] => 0
[11,2,2] => [6,2,2,2,2,1] => 5
[9,4,1,1] => [4,4,4,1,1,1] => 3
[9,3,2,1] => [3,3,3,2,2,2] => 3
[8,7] => [4,4,4,3] => 3
[7,5,2,1] => [4,3,3,3,2] => 2
[7,3,3,2] => [5,3,2,2,2,1] => 3
[6,6,3] => [6,5,4] => 2
[6,5,4] => [6,6,3] => 2
[6,4,3,2] => [8,5,2] => 2
[6,3,2,1,1,1,1] => [6,5,1,1,1,1] => 1
[6,2,2,2,2,1] => [11,2,2] => 2
[5,5,5] => [5,4,3,2,1] => 2
[5,5,4,1] => [5,4,3,3] => 1
[5,5,2,2,1] => [6,3,3,3] => 1
[5,5,2,1,1,1] => [5,4,3,1,1,1] => 1
[5,4,3,2,1] => [5,5,5] => 0
[4,4,4,3] => [8,7] => 1
[4,4,4,2,1] => [9,6] => 1
[4,3,3,2,2,1] => [7,5,3] => 0
[4,3,2,2,2,1,1] => [9,5,1] => 0
[3,3,3,3,3] => [3,3,3,3,2,1] => 1
[3,3,3,3,2,1] => [3,3,3,3,3] => 0
[3,2,2,2,2,2,2] => [14,1] => 1
[2,2,2,2,2,2,2,1] => [15] => 0
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0
[16] => [2,2,2,2,2,2,2,2] => 8
[13,1,1,1] => [2,2,2,2,2,2,1,1,1,1] => 6
[10,6] => [4,4,4,2,2] => 5
[10,3,3] => [5,4,2,2,2,1] => 4
[10,3,2,1] => [5,5,2,2,2] => 3
[9,6,1] => [4,4,3,3,2] => 3
[9,1,1,1,1,1,1,1] => [2,2,2,2,1,1,1,1,1,1,1,1] => 4
[8,8] => [4,4,4,4] => 4
[7,6,3] => [5,4,4,3] => 2
[7,6,2,1] => [6,4,3,3] => 2
[7,3,3,2,1] => [3,3,3,3,2,2] => 2
[6,6,4] => [6,6,4] => 3
[6,5,4,1] => [6,5,5] => 1
[6,4,4,2] => [8,6,2] => 3
[5,5,5,1] => [5,4,3,2,1,1] => 2
[5,5,3,3] => [4,3,3,3,2,1] => 2
[5,5,3,2,1] => [4,3,3,3,3] => 1
[5,4,4,3] => [7,6,3] => 1
[5,4,4,2,1] => [8,5,3] => 1
[4,4,4,4] => [8,8] => 2
[4,4,4,2,2] => [10,6] => 2
[4,4,2,2,2,2] => [12,4] => 2
[4,3,3,3,2,1] => [5,5,3,3] => 0
[4,3,3,2,2,1,1] => [7,5,3,1] => 0
[2,2,2,2,2,2,2,2] => [16] => 1
[2,2,2,2,2,2,2,1,1] => [15,1] => 0
[16,1] => [3,2,2,2,2,2,2,2] => 7
[11,6] => [4,4,4,2,2,1] => 5
[10,7] => [4,4,4,3,2] => 4
[10,4,3] => [6,5,2,2,2] => 4
[9,6,1,1] => [4,4,4,3,1,1] => 3
[9,4,2,1,1] => [6,4,4,1,1,1] => 3
[8,8,1] => [5,4,4,4] => 3
[8,7,2] => [6,4,4,3] => 3
[8,3,3,3] => [5,5,2,2,2,1] => 3
[7,7,2,1] => [4,4,3,3,3] => 2
[7,5,5] => [5,4,3,2,2,1] => 3
[7,5,2,2,1] => [6,3,3,3,2] => 2
[6,5,4,2] => [8,6,3] => 2
[6,5,3,2,1] => [6,5,3,3] => 1
[6,3,3,2,2,1] => [7,5,3,2] => 1
[5,5,4,3] => [5,5,4,3] => 1
[5,4,4,4] => [8,8,1] => 2
[5,4,3,3,1,1] => [5,5,4,1,1,1] => 1
[5,3,3,3,2,1] => [3,3,3,3,3,2] => 1
[5,3,3,2,2,1,1] => [6,3,3,3,1,1] => 1
[4,4,4,4,1] => [9,8] => 1
[3,3,3,3,3,2] => [5,3,3,3,2,1] => 1
[2,2,2,2,2,2,2,2,1] => [17] => 0
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Description
The number of even parts of a partition.
Map
2-conjugate
Description
Return a partition with the same number of odd parts and number of even parts interchanged with the number of cells with zero leg and odd arm length.
This is a special case of an involution that preserves the sequence of non-zero remainders of the parts under division by $s$ and interchanges the number of parts divisible by $s$ and the number of cells with zero leg length and arm length congruent to $s-1$ modulo $s$.
In particular, for $s=1$ the involution is conjugation, hence the name.