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] => [3,1,2] => ([(1,2)],3) => [6] => 1
[2,1,3] => [2,1,3] => ([(0,2),(1,2)],3) => [3,2] => 1
[2,3,1] => [2,3,1] => ([(1,2)],3) => [6] => 1
[3,1,2] => [1,3,2] => ([(0,1),(0,2)],3) => [3,2] => 1
[3,2,1] => [3,2,1] => ([],3) => [2,2,2,2] => 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => [5] => 1
[1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4) => [4,4] => 1
[1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => [7] => 1
[1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => [5,3] => 1
[1,4,2,3] => [1,4,2,3] => ([(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] => [4,2,1,3] => ([(1,3),(2,3)],4) => [6,2,2] => 1
[2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => [7] => 1
[2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4) => [4,4] => 1
[2,4,1,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => [3,2,2] => 1
[3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => [4,2] => 1
[3,1,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4) => [3,3,3] => 1
[3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => [3,2,2,2] => 1
[3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4) => [6,2,2] => 1
[3,4,1,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => [7] => 1
[4,1,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => [4,2] => 1
[4,1,3,2] => [4,1,3,2] => ([(1,2),(1,3)],4) => [6,2,2] => 1
[4,2,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => [5,3] => 1
[4,2,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4) => [6,2,2] => 1
[4,3,1,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => [3,2,2,2] => 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] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => [10] => 1
[1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => [5,4] => 1
[1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5) => [10] => 1
[1,2,5,3,4] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5) => [5,4] => 1
[1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => [8] => 1
[1,3,4,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [5,4] => 1
[1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5) => [10] => 1
[1,3,5,2,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [7,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [8] => 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
[1,5,2,3,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5) => [8] => 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => [5,2] => 1
[2,1,5,3,4] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => [10] => 1
[2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => [8] => 1
[2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => [5,4] => 1
[2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => [10] => 1
[2,3,5,1,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [7,2] => 1
[2,4,1,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [4,2,2] => 1
[2,4,5,1,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [7,2] => 1
[2,5,1,3,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => [4,2,2] => 1
[3,1,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [5,2] => 1
[3,1,4,2,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => [4,3,3] => 1
[3,1,5,2,4] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [10] => 1
[3,2,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => [4,2,2,2] => 1
[3,4,1,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [8] => 1
[3,4,5,1,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => [5,4] => 1
[3,5,1,2,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [4,2,2] => 1
[4,1,2,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [5,2] => 1
[4,1,5,2,3] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5) => [4,3,3] => 1
[4,2,1,3,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [5,4] => 1
[4,2,5,1,3] => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [10] => 1
[4,3,1,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [4,2,2,2] => 1
[4,5,1,2,3] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => [8] => 1
[5,1,2,3,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => [5,2] => 1
[5,2,1,3,4] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [7,2] => 1
[5,2,3,1,4] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5) => [10] => 1
[5,3,1,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [5,4] => 1
[5,4,1,2,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => [4,2,2,2] => 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
[1,2,4,3,5,6] => [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [6,4] => 1
[1,2,5,3,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => [6,4] => 1
[1,2,6,3,4,5] => [1,2,6,3,4,5] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => [6,4] => 1
[1,3,2,4,5,6] => [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [9] => 1
[1,3,4,2,5,6] => [3,1,4,2,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => [5,5] => 1
[1,3,5,2,4,6] => [3,1,2,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [8,2] => 1
[1,3,6,2,4,5] => [3,1,2,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6) => [8,2] => 1
[1,4,2,3,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => [9] => 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
[1,4,6,2,3,5] => [1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6) => [8,2] => 1
[1,5,2,3,4,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => [9] => 1
[1,5,6,2,3,4] => [1,2,5,3,6,4] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [5,5] => 1
[1,6,2,3,4,5] => [1,2,3,6,4,5] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => [9] => 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
[2,3,1,4,5,6] => [2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [9] => 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [6,4] => 1
[2,3,5,1,4,6] => [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => [8,2] => 1
[2,3,6,1,4,5] => [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6) => [8,2] => 1
[2,4,1,3,5,6] => [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
[2,4,5,1,3,6] => [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6) => [8,2] => 1
[2,4,6,1,3,5] => [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
[2,5,1,3,4,6] => [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6) => [5,2,2] => 1
[2,5,6,1,3,4] => [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6) => [8,2] => 1
[2,6,1,3,4,5] => [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6) => [5,2,2] => 1
[3,1,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => [6,2] => 1
[3,4,1,2,5,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => [9] => 1
[3,4,5,1,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => [6,4] => 1
[3,4,6,1,2,5] => [1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6) => [8,2] => 1
[3,5,1,2,4,6] => [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
[3,5,6,1,2,4] => [1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => [8,2] => 1
[3,6,1,2,4,5] => [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6) => [5,2,2] => 1
[4,1,2,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => [6,2] => 1
>>> Load all 140 entries. <<<
[4,2,1,3,5,6] => [2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => [5,5] => 1
[4,5,1,2,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => [9] => 1
[4,5,6,1,2,3] => [1,2,4,5,6,3] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6) => [6,4] => 1
[4,6,1,2,3,5] => [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
[5,1,2,3,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => [6,2] => 1
[5,2,1,3,4,6] => [2,1,5,3,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6) => [8,2] => 1
[5,3,1,2,4,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => [5,5] => 1
[5,6,1,2,3,4] => [1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6) => [9] => 1
[6,1,2,3,4,5] => [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6) => [6,2] => 1
[6,2,1,3,4,5] => [2,1,3,6,4,5] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6) => [8,2] => 1
[6,3,1,2,4,5] => [1,3,2,6,4,5] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => [8,2] => 1
[6,4,1,2,3,5] => [1,2,4,6,3,5] => ([(0,4),(2,5),(3,1),(3,5),(4,2),(4,3)],6) => [5,5] => 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
[1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [10] => 1
[1,4,2,3,5,6,7] => [1,4,2,3,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => [10] => 1
[1,5,2,3,4,6,7] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => [10] => 1
[1,6,2,3,4,5,7] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => [10] => 1
[1,7,2,3,4,5,6] => [1,2,3,4,7,5,6] => ([(0,5),(3,6),(4,1),(5,3),(6,2),(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
[2,3,1,4,5,6,7] => [2,3,1,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7) => [10] => 1
[2,4,1,3,5,6,7] => [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
[2,5,1,3,4,6,7] => [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
[2,6,1,3,4,5,7] => [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
[2,7,1,3,4,5,6] => [2,1,3,4,5,7,6] => ([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7) => [6,2,2] => 1
[3,1,2,4,5,6,7] => [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
[3,4,1,2,5,6,7] => [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => [10] => 1
[3,5,1,2,4,6,7] => [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
[3,6,1,2,4,5,7] => [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
[3,7,1,2,4,5,6] => [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
[4,1,2,3,5,6,7] => [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
[4,5,1,2,3,6,7] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => [10] => 1
[4,6,1,2,3,5,7] => [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
[4,7,1,2,3,5,6] => [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
[5,1,2,3,4,6,7] => [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
[5,6,1,2,3,4,7] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => [10] => 1
[5,7,1,2,3,4,6] => [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
[6,1,2,3,4,5,7] => [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
[6,7,1,2,3,4,5] => [1,2,3,4,6,7,5] => ([(0,5),(3,6),(4,1),(5,3),(6,2),(6,4)],7) => [10] => 1
[7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7) => [7,2] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The smallest positive integer that does not appear twice in the partition.
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
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
rowmotion cycle type
Description
The cycle type of rowmotion on the order ideals of a poset.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
  • If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
  • If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).