Identifier
Values
[1] => [1] => ([],1) => [2] => 1
[1,2] => [1,2] => ([(0,1)],2) => [3] => 1
[2,1] => [2,1] => ([],2) => [2,2] => 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => [4] => 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2)],3) => [3,2] => 1
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3) => [3,2] => 1
[2,3,1] => [3,1,2] => ([(1,2)],3) => [6] => 1
[3,1,2] => [3,2,1] => ([],3) => [2,2,2,2] => 1
[3,2,1] => [2,3,1] => ([(1,2)],3) => [6] => 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => [5] => 1
[1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => [4,2] => 1
[1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => [4,2] => 1
[1,3,4,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4) => [7] => 1
[1,4,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => [3,2,2,2] => 1
[1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => [7] => 1
[2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => [4,2] => 1
[2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => [3,2,2] => 1
[2,3,1,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => [7] => 1
[2,3,4,1] => [4,1,2,3] => ([(1,2),(2,3)],4) => [4,4] => 1
[2,4,3,1] => [3,4,1,2] => ([(0,3),(1,2)],4) => [3,3,3] => 1
[3,1,2,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => [3,2,2,2] => 1
[3,1,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4) => [6,2,2] => 1
[3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => [7] => 1
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => [5,3] => 1
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => [5,3] => 1
[3,4,2,1] => [4,1,3,2] => ([(1,2),(1,3)],4) => [6,2,2] => 1
[4,2,1,3] => [2,4,3,1] => ([(1,2),(1,3)],4) => [6,2,2] => 1
[4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4) => [4,4] => 1
[4,3,2,1] => [3,2,4,1] => ([(1,3),(2,3)],4) => [6,2,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => [6] => 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => [5,2] => 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [5,2] => 1
[1,2,4,5,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5) => [8] => 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => [4,2,2,2] => 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => [8] => 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [5,2] => 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [4,2,2] => 1
[1,3,4,2,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [8] => 1
[1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5) => [5,4] => 1
[1,3,5,4,2] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5) => [4,3,3] => 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [4,2,2,2] => 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [8] => 1
[1,4,3,5,2] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [5,4] => 1
[1,4,5,2,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [5,4] => 1
[1,5,3,4,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => [5,4] => 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => [5,2] => 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => [4,2,2] => 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [4,2,2] => 1
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [7,2] => 1
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [7,2] => 1
[2,3,1,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => [8] => 1
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [7,2] => 1
[2,3,4,1,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => [5,4] => 1
[2,3,4,5,1] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => [10] => 1
[2,4,3,1,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => [4,3,3] => 1
[2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5) => [10] => 1
[3,1,2,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => [4,2,2,2] => 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => [8] => 1
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [7,2] => 1
[3,2,4,1,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [5,4] => 1
[3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => [10] => 1
[3,4,1,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [5,4] => 1
[3,4,1,5,2] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [10] => 1
[3,5,1,4,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5) => [10] => 1
[4,2,3,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => [5,4] => 1
[4,2,3,5,1] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5) => [10] => 1
[4,2,5,1,3] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [10] => 1
[5,2,3,4,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => [10] => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [7] => 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6) => [6,2] => 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => [6,2] => 1
[1,2,3,5,6,4] => [1,2,3,6,4,5] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => [9] => 1
[1,2,3,6,5,4] => [1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => [9] => 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => [6,2] => 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6) => [5,2,2] => 1
[1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => [9] => 1
[1,2,4,5,6,3] => [1,2,6,3,4,5] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => [6,4] => 1
[1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => [9] => 1
[1,2,5,4,6,3] => [1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [5,5] => 1
[1,2,5,6,3,4] => [1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [5,5] => 1
[1,2,6,4,5,3] => [1,2,4,5,6,3] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => [6,4] => 1
[1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => [6,2] => 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6) => [5,2,2] => 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [5,2,2] => 1
[1,3,2,5,6,4] => [1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => [8,2] => 1
[1,3,2,6,5,4] => [1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => [8,2] => 1
[1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => [9] => 1
[1,3,4,2,6,5] => [1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6) => [8,2] => 1
[1,3,4,5,2,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => [6,4] => 1
[1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => [9] => 1
[1,4,3,2,6,5] => [1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6) => [8,2] => 1
[1,4,3,5,2,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [5,5] => 1
[1,4,5,2,3,6] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [5,5] => 1
[1,5,3,4,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => [6,4] => 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [6,2] => 1
[2,1,3,4,6,5] => [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6) => [5,2,2] => 1
[2,1,3,5,4,6] => [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6) => [5,2,2] => 1
[2,1,3,5,6,4] => [2,1,3,6,4,5] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6) => [8,2] => 1
[2,1,3,6,5,4] => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6) => [8,2] => 1
[2,1,4,3,5,6] => [2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6) => [5,2,2] => 1
[2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6) => [4,2,2,2] => 1
>>> Load all 140 entries. <<<
[2,1,4,5,3,6] => [2,1,5,3,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6) => [8,2] => 1
[2,1,5,4,3,6] => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6) => [8,2] => 1
[2,3,1,4,5,6] => [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [9] => 1
[2,3,1,4,6,5] => [3,1,2,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6) => [8,2] => 1
[2,3,1,5,4,6] => [3,1,2,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [8,2] => 1
[2,3,4,1,5,6] => [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [6,4] => 1
[3,2,1,4,5,6] => [2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [9] => 1
[3,2,1,4,6,5] => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6) => [8,2] => 1
[3,2,1,5,4,6] => [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [8,2] => 1
[3,2,4,1,5,6] => [2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => [5,5] => 1
[3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => [5,5] => 1
[4,2,3,1,5,6] => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [6,4] => 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => [8] => 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7) => [7,2] => 1
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => [7,2] => 1
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7) => [10] => 1
[1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7) => [10] => 1
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => [7,2] => 1
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7) => [6,2,2] => 1
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => [10] => 1
[1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => [10] => 1
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => [7,2] => 1
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7) => [6,2,2] => 1
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7) => [6,2,2] => 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => [10] => 1
[1,2,5,4,3,6,7] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => [10] => 1
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => [7,2] => 1
[1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7) => [6,2,2] => 1
[1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => [6,2,2] => 1
[1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7) => [6,2,2] => 1
[1,3,4,2,5,6,7] => [1,4,2,3,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => [10] => 1
[1,4,3,2,5,6,7] => [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => [10] => 1
[2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7) => [7,2] => 1
[2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7) => [6,2,2] => 1
[2,1,3,4,6,5,7] => [2,1,3,4,6,5,7] => ([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7) => [6,2,2] => 1
[2,1,3,5,4,6,7] => [2,1,3,5,4,6,7] => ([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7) => [6,2,2] => 1
[2,1,4,3,5,6,7] => [2,1,4,3,5,6,7] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7) => [6,2,2] => 1
[2,3,1,4,5,6,7] => [3,1,2,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [10] => 1
[3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [10] => 1
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Description
The smallest positive integer that does not appear twice in the partition.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Map
rowmotion cycle type
Description
The cycle type of rowmotion on the order ideals of a poset.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$