Processing math: 100%

Identifier
Values
([],1) => [2] => 2
([],2) => [2,2] => 2
([(0,1)],2) => [3] => 3
([],3) => [2,2,2,2] => 2
([(1,2)],3) => [6] => 6
([(0,1),(0,2)],3) => [3,2] => 3
([(0,2),(2,1)],3) => [4] => 4
([(0,2),(1,2)],3) => [3,2] => 3
([(2,3)],4) => [6,6] => 6
([(1,2),(1,3)],4) => [6,2,2] => 6
([(0,1),(0,2),(0,3)],4) => [3,2,2,2] => 3
([(0,2),(0,3),(3,1)],4) => [7] => 7
([(0,1),(0,2),(1,3),(2,3)],4) => [4,2] => 4
([(1,2),(2,3)],4) => [4,4] => 4
([(0,3),(3,1),(3,2)],4) => [4,2] => 4
([(1,3),(2,3)],4) => [6,2,2] => 6
([(0,3),(1,3),(3,2)],4) => [4,2] => 4
([(0,3),(1,3),(2,3)],4) => [3,2,2,2] => 3
([(0,3),(1,2)],4) => [3,3,3] => 3
([(0,3),(1,2),(1,3)],4) => [5,3] => 5
([(0,2),(0,3),(1,2),(1,3)],4) => [3,2,2] => 3
([(0,3),(2,1),(3,2)],4) => [5] => 5
([(0,3),(1,2),(2,3)],4) => [7] => 7
([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => [7,2,2] => 7
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [4,2,2,2] => 4
([(0,3),(0,4),(4,1),(4,2)],5) => [7,2,2] => 7
([(1,2),(1,3),(2,4),(3,4)],5) => [4,4,2,2] => 4
([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [5,2] => 5
([(0,3),(0,4),(3,2),(4,1)],5) => [4,3,3] => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [5,4] => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [4,2,2] => 4
([(1,4),(4,2),(4,3)],5) => [4,4,2,2] => 4
([(0,4),(4,1),(4,2),(4,3)],5) => [4,2,2,2] => 4
([(1,4),(2,4),(4,3)],5) => [4,4,2,2] => 4
([(0,4),(1,4),(4,2),(4,3)],5) => [4,2,2] => 4
([(0,4),(1,4),(2,4),(4,3)],5) => [4,2,2,2] => 4
([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [5,3,2,2] => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [3,2,2,2,2] => 3
([(0,4),(1,4),(2,3),(4,2)],5) => [5,2] => 5
([(0,4),(1,3),(2,3),(3,4)],5) => [7,2,2] => 7
([(0,4),(1,2),(1,4),(2,3)],5) => [8,3] => 8
([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [5,4] => 5
([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [7,2] => 7
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [4,2,2] => 4
([(0,4),(1,2),(1,4),(4,3)],5) => [10] => 10
([(0,2),(0,4),(3,1),(4,3)],5) => [5,4] => 5
([(0,4),(1,2),(1,3),(3,4)],5) => [10,2] => 10
([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [8] => 8
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [7,2,2] => 7
([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [5,3,2,2] => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [3,2,2,2,2] => 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [10] => 10
([(0,3),(1,2),(1,4),(3,4)],5) => [8,3] => 8
([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [7,2] => 7
([(1,4),(3,2),(4,3)],5) => [10] => 10
([(0,3),(3,4),(4,1),(4,2)],5) => [5,2] => 5
([(0,4),(1,2),(2,4),(4,3)],5) => [8] => 8
([(0,3),(1,4),(4,2)],5) => [12] => 12
([(0,4),(3,2),(4,1),(4,3)],5) => [8] => 8
([(0,4),(1,2),(2,3),(2,4)],5) => [10] => 10
([(0,4),(2,3),(3,1),(4,2)],5) => [6] => 6
([(0,3),(1,2),(2,4),(3,4)],5) => [4,3,3] => 4
([(0,4),(1,2),(2,3),(3,4)],5) => [5,4] => 5
([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [5,2] => 5
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => [5,2,2,2] => 5
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => [4,2,2,2,2] => 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => [8,2,2] => 8
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6) => [5,2,2] => 5
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => [8,2] => 8
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [5,2,2] => 5
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6) => [11] => 11
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => [4,2,2,2,2] => 4
([(0,4),(4,5),(5,1),(5,2),(5,3)],6) => [5,2,2,2] => 5
([(0,5),(1,5),(5,2),(5,3),(5,4)],6) => [4,2,2,2,2] => 4
([(0,5),(1,5),(2,5),(5,3),(5,4)],6) => [4,2,2,2,2] => 4
([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => [5,2,2,2] => 5
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [4,2,2,2,2] => 4
([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [8,2,2] => 8
([(0,5),(1,5),(4,2),(5,3),(5,4)],6) => [8,2] => 8
([(0,5),(1,5),(4,2),(4,3),(5,4)],6) => [5,2,2] => 5
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => [5,3,3] => 5
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => [8,2,2] => 8
([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [6,2] => 6
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => [8,2,2] => 8
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => [5,2,2,2] => 5
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6) => [5,2,2] => 5
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6) => [5,4,2] => 5
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => [5,5] => 5
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6) => [8,4] => 8
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6) => [4,3,3,2] => 4
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6) => [5,4,2] => 5
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6) => [4,2,2,2] => 4
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6) => [5,4,2] => 5
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6) => [5,2,2] => 5
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6) => [8,2] => 8
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6) => [9,3] => 9
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [5,5] => 5
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6) => [9,3] => 9
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6) => [11] => 11
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => [9] => 9
([(0,5),(1,2),(1,3),(2,5),(3,5),(5,4)],6) => [8,2,2] => 8
>>> Load all 225 entries. <<<
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => [6,4] => 6
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => [6,2] => 6
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6) => [4,2,2,2,2] => 4
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6) => [7,5] => 7
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6) => [11] => 11
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6) => [8,4] => 8
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6) => [8,2] => 8
([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6) => [11] => 11
([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6) => [11] => 11
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [8,2] => 8
([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6) => [9,3] => 9
([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => [11] => 11
([(0,5),(4,2),(4,3),(5,1),(5,4)],6) => [8,2,2] => 8
([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6) => [7,5] => 7
([(0,5),(1,2),(2,5),(5,3),(5,4)],6) => [8,2] => 8
([(0,5),(1,4),(4,2),(4,5),(5,3)],6) => [6,6] => 6
([(1,5),(3,4),(4,2),(5,3)],6) => [6,6] => 6
([(0,4),(3,5),(4,3),(5,1),(5,2)],6) => [6,2] => 6
([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => [5,3,3] => 5
([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => [6,4] => 6
([(0,3),(1,2),(2,4),(2,5),(3,4),(3,5)],6) => [4,3,3,2] => 4
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => [6,2] => 6
([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [6,4] => 6
([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => [9,3] => 9
([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6) => [11] => 11
([(0,5),(1,3),(1,5),(4,2),(5,4)],6) => [7,5] => 7
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6) => [5,4,2] => 5
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6) => [5,2,2] => 5
([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [5,5] => 5
([(0,5),(3,2),(4,1),(5,3),(5,4)],6) => [5,3,3] => 5
([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6) => [11] => 11
([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => [9] => 9
([(0,5),(1,3),(3,4),(4,2),(4,5)],6) => [7,5] => 7
([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [7] => 7
([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [9] => 9
([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [11] => 11
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => [6,2] => 6
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => [9] => 9
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7) => [6,2,2] => 6
([(0,2),(0,3),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1),(6,5)],7) => [5,5,2] => 5
([(0,1),(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7) => [5,2,2,2] => 5
([(0,4),(0,5),(1,6),(4,6),(5,1),(6,2),(6,3)],7) => [9,2] => 9
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7) => [6,2,2,2] => 6
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => [6,2,2,2] => 6
([(0,6),(1,6),(4,3),(5,2),(5,4),(6,5)],7) => [9,2] => 9
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => [12] => 12
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7) => [6,2,2] => 6
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => [6,2,2] => 6
([(0,6),(1,6),(4,2),(5,4),(6,3),(6,5)],7) => [6,4,2] => 6
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7) => [6,2,2] => 6
([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7) => [12] => 12
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => [7,2] => 7
([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7) => [7,2] => 7
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7) => [6,2,2] => 6
([(0,6),(1,6),(3,5),(4,2),(4,5),(6,3),(6,4)],7) => [5,5,2] => 5
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(6,2),(6,3)],7) => [5,2,2,2] => 5
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,3),(5,4),(6,2),(6,4)],7) => [5,5,2] => 5
([(0,5),(0,6),(1,5),(1,6),(3,2),(4,2),(5,3),(5,4),(6,3),(6,4)],7) => [5,2,2,2] => 5
([(0,6),(1,6),(2,5),(3,5),(4,3),(6,2),(6,4)],7) => [9,2] => 9
([(0,6),(1,2),(2,6),(3,5),(4,5),(6,3),(6,4)],7) => [9,2] => 9
([(0,5),(1,4),(1,5),(4,6),(5,6),(6,2),(6,3)],7) => [5,5,2] => 5
([(0,5),(0,6),(1,5),(1,6),(4,2),(5,3),(5,4),(6,3),(6,4)],7) => [8,2,2] => 8
([(0,5),(0,6),(1,5),(1,6),(4,2),(4,3),(5,4),(6,4)],7) => [5,2,2,2] => 5
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(5,2),(6,3),(6,4)],7) => [8,2,2] => 8
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7) => [6,2,2] => 6
([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,3),(6,4)],7) => [9,2] => 9
([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(5,2),(6,4)],7) => [6,4,2] => 6
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => [6,2,2,2] => 6
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7) => [5,5,2] => 5
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => [6,5] => 6
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5),(4,6),(5,6)],7) => [12] => 12
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => [7,4] => 7
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7) => [6,4,2] => 6
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => [7,2] => 7
([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7) => [9,2] => 9
([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7) => [6,4,2] => 6
([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7) => [9,2] => 9
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => [6,3,3] => 6
([(0,3),(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(6,1)],7) => [12] => 12
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(4,6),(5,6)],7) => [9,2] => 9
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => [12] => 12
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7) => [6,2,2] => 6
([(0,2),(1,5),(1,6),(2,5),(2,6),(5,3),(5,4),(6,3),(6,4)],7) => [8,2,2] => 8
([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7) => [12] => 12
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => [6,5] => 6
([(0,2),(1,5),(1,6),(2,3),(3,5),(3,6),(5,4),(6,4)],7) => [6,4,2] => 6
([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7) => [9,2] => 9
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7) => [6,2,2] => 6
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7) => [6,2,2] => 6
([(0,5),(1,4),(3,6),(4,3),(4,5),(5,6),(6,2)],7) => [12] => 12
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => [6,2,2,2] => 6
([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7) => [6,2,2,2] => 6
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => [10] => 10
([(0,6),(1,5),(2,6),(5,2),(6,3),(6,4)],7) => [6,4,2] => 6
([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7) => [12] => 12
([(0,6),(1,2),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4),(6,5)],7) => [5,5,2] => 5
([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7) => [6,5] => 6
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => [12] => 12
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => [10] => 10
([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7) => [12] => 12
([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7) => [12] => 12
([(0,3),(1,5),(1,6),(3,5),(3,6),(4,2),(5,4),(6,4)],7) => [9,2] => 9
([(0,3),(1,4),(1,6),(2,5),(3,4),(3,6),(4,2),(6,5)],7) => [12] => 12
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7) => [9,2] => 9
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7) => [6,2,2] => 6
([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7) => [6,5] => 6
([(0,6),(1,4),(4,6),(5,2),(5,3),(6,5)],7) => [9,2] => 9
([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7) => [7,2] => 7
([(0,5),(3,4),(4,1),(5,6),(6,2),(6,3)],7) => [7,4] => 7
([(0,6),(3,5),(4,3),(5,1),(6,2),(6,4)],7) => [12] => 12
([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7) => [12] => 12
([(0,6),(1,3),(3,6),(5,2),(6,4),(6,5)],7) => [12] => 12
([(0,5),(3,2),(4,1),(5,6),(6,3),(6,4)],7) => [6,3,3] => 6
([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7) => [10] => 10
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [8] => 8
([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [10] => 10
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => [10] => 10
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => [6,3,3] => 6
([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7) => [6,3,3] => 6
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => [7,2] => 7
([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7) => [7,4] => 7
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => [7,2] => 7
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => [7,4] => 7
([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8) => [7,5] => 7
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Description
Dyson's crank of a partition.
Let λ be a partition and let o(λ) be the number of parts that are equal to 1 (St000475The number of parts equal to 1 in a partition.), and let μ(λ) be the number of parts that are strictly larger than o(λ) (St000473The number of parts of a partition that are strictly bigger than the number of ones.). Dyson's crank is then defined as
crank(λ)={ largest part of λo(λ)=0μ(λ)−o(λ)o(λ)>0.
Map
rowmotion cycle type
Description
The cycle type of rowmotion on the order ideals of a poset.