Identifier
Values
[1,1] => [1] => [1,0] => [1,0] => 0
[2,1] => [1] => [1,0] => [1,0] => 0
[1,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[3,1] => [1] => [1,0] => [1,0] => 0
[2,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[2,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[4,1] => [1] => [1,0] => [1,0] => 0
[3,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[3,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[2,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[5,1] => [1] => [1,0] => [1,0] => 0
[4,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[4,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[3,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
[3,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[2,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[6,1] => [1] => [1,0] => [1,0] => 0
[5,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[5,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[4,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
[4,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[3,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[2,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[7,1] => [1] => [1,0] => [1,0] => 0
[6,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[6,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[5,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
[5,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[4,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[3,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => 2
[3,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[2,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => 3
[8,1] => [1] => [1,0] => [1,0] => 0
[7,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[7,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[6,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
[6,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[5,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[4,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => 2
[4,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[3,3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
[3,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => 3
[2,2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[9,1] => [1] => [1,0] => [1,0] => 0
[8,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[8,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[7,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
[7,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[6,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[5,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => 2
[5,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[4,3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
[4,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => 3
[3,3,2,2] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[3,2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[10,1] => [1] => [1,0] => [1,0] => 0
[9,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[9,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[8,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
[8,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[7,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[6,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => 2
[6,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[5,3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
[5,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => 3
[4,3,2,2] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[4,2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[3,3,2,2,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[11,1] => [1] => [1,0] => [1,0] => 0
[10,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[10,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[9,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
[9,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[8,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[7,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => 2
[7,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[6,3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
[6,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => 3
[5,3,2,2] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[5,2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[4,3,2,2,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[3,3,3,3] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 4
[12,1] => [1] => [1,0] => [1,0] => 0
[11,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[11,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[10,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
[10,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[9,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[8,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => 2
[8,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[7,3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
[7,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => 3
[6,3,2,2] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[6,2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[5,3,2,2,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[4,3,3,3] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 4
[3,3,3,3,1] => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[13,1] => [1] => [1,0] => [1,0] => 0
[12,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[12,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[11,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
>>> Load all 183 entries. <<<
[11,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[10,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[9,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => 2
[9,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[8,3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
[8,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => 3
[7,3,2,2] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[7,2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[6,3,2,2,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[5,3,3,3] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 4
[4,4,3,3] => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[4,3,3,3,1] => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[3,3,3,3,2] => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 7
[14,1] => [1] => [1,0] => [1,0] => 0
[13,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[13,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[12,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
[12,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[11,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[10,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => 2
[10,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[9,3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
[9,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => 3
[8,3,2,2] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[8,2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[7,3,2,2,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[6,3,3,3] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 4
[5,4,3,3] => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[5,3,3,3,1] => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[4,4,3,3,1] => [4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5
[4,3,3,3,2] => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 7
[3,3,3,3,3] => [3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[15,1] => [1] => [1,0] => [1,0] => 0
[14,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[14,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[13,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
[13,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[12,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[11,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => 2
[11,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[10,3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
[10,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => 3
[9,3,2,2] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[9,2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[8,3,2,2,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[7,3,3,3] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 4
[6,4,3,3] => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[6,3,3,3,1] => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[5,4,3,3,1] => [4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5
[5,3,3,3,2] => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 7
[4,4,3,3,2] => [4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,1,0,0,0] => 7
[4,3,3,3,3] => [3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[3,3,3,3,3,1] => [3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[16,1] => [1] => [1,0] => [1,0] => 0
[15,2] => [2] => [1,0,1,0] => [1,1,0,0] => 0
[15,1,1] => [1,1] => [1,1,0,0] => [1,0,1,0] => 1
[14,3] => [3] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 2
[14,2,1] => [2,1] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[13,2,2] => [2,2] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 2
[12,3,2] => [3,2] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => 2
[12,2,2,1] => [2,2,1] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 3
[11,3,2,1] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 3
[11,2,2,2] => [2,2,2] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => 3
[10,3,2,2] => [3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 3
[10,2,2,2,1] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[9,3,2,2,1] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[8,3,3,3] => [3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 4
[7,4,3,3] => [4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[7,3,3,3,1] => [3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[6,4,3,3,1] => [4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5
[6,3,3,3,2] => [3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 7
[5,4,3,3,2] => [4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,1,0,0,0] => 7
[5,3,3,3,3] => [3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[4,4,3,3,3] => [4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[4,3,3,3,3,1] => [3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6
[3,3,3,3,3,2] => [3,3,3,3,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => 8
[6,4,3,3,2] => [4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,1,0,0,0] => 7
[4,3,3,3,3,2] => [3,3,3,3,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => 8
[5,4,3,3,3] => [4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[7,3,3,3,3] => [3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[4,4,4,4,4] => [4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[5,4,4,4,4] => [4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
search for individual values
searching the database for the individual values of this statistic
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
first row removal
Description
Removes the first entry of an integer partition
Map
promotion
Description
The promotion of the two-row standard Young tableau of a Dyck path.
Dyck paths of semilength $n$ are in bijection with standard Young tableaux of shape $(n^2)$, see Mp00033to two-row standard tableau.
This map is the bijection on such standard Young tableaux given by Schützenberger's promotion. For definitions and details, see [1] and the references therein.