- FindStat

Identifier

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000180: Posets ⟶ ℤ

Values

[1,0] => [1] => [1] => ([],1) => 2

[1,0,1,0] => [2,1] => [1,2] => ([(0,1)],2) => 4

[1,1,0,0] => [1,2] => [2,1] => ([],2) => 3

[1,0,1,0,1,0] => [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3) => 6

[1,0,1,1,0,0] => [2,1,3] => [3,1,2] => ([(1,2)],3) => 5

[1,1,0,0,1,0] => [1,3,2] => [2,3,1] => ([(1,2)],3) => 5

[1,1,0,1,0,0] => [3,1,2] => [2,1,3] => ([(0,2),(1,2)],3) => 6

[1,1,1,0,0,0] => [1,2,3] => [3,2,1] => ([],3) => 4

[1,0,1,0,1,0,1,0] => [2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 8

[1,0,1,0,1,1,0,0] => [2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4) => 7

[1,0,1,1,0,0,1,0] => [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4) => 7

[1,0,1,1,0,1,0,0] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 8

[1,0,1,1,1,0,0,0] => [2,1,3,4] => [4,3,1,2] => ([(2,3)],4) => 6

[1,1,0,0,1,0,1,0] => [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4) => 7

[1,1,0,0,1,1,0,0] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4) => 6

[1,1,0,1,0,0,1,0] => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => 8

[1,1,0,1,0,1,0,0] => [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 9

[1,1,0,1,1,0,0,0] => [3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4) => 7

[1,1,1,0,0,0,1,0] => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4) => 6

[1,1,1,0,0,1,0,0] => [1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4) => 7

[1,1,1,0,1,0,0,0] => [4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 8

[1,1,1,1,0,0,0,0] => [1,2,3,4] => [4,3,2,1] => ([],4) => 5

[1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 10

[1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [5,1,4,3,2] => ([(1,2),(1,3),(1,4)],5) => 9

[1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5) => 9

[1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5) => 10

[1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [5,4,1,3,2] => ([(2,3),(2,4)],5) => 8

[1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5) => 9

[1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [5,3,4,1,2] => ([(1,4),(2,3)],5) => 8

[1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 10

[1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 11

[1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5) => 9

[1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [4,5,3,1,2] => ([(1,4),(2,3)],5) => 8

[1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5) => 9

[1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5) => 10

[1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [5,4,3,1,2] => ([(3,4)],5) => 7

[1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5) => 9

[1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5) => 8

[1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5) => 8

[1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5) => 9

[1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5) => 7

[1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 10

[1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5) => 9

[1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 11

[1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 12

[1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5) => 10

[1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5) => 9

[1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5) => 10

[1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 11

[1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 8

[1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5) => 8

[1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5) => 7

[1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5) => 9

[1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5) => 10

[1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 8

[1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5) => 10

[1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 11

[1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 12

[1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5) => 9

[1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5) => 7

[1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5) => 8

[1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5) => 9

[1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 10

[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5) => 6

[1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [1,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => 12

[1,0,1,0,1,0,1,0,1,1,0,0] => [2,3,4,5,1,6] => [6,1,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5)],6) => 11

[1,0,1,0,1,0,1,1,1,0,0,0] => [2,3,4,1,5,6] => [6,5,1,4,3,2] => ([(2,3),(2,4),(2,5)],6) => 10

[1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [4,6,5,1,3,2] => ([(0,4),(0,5),(1,2),(1,3)],6) => 11

[1,0,1,0,1,1,1,1,0,0,0,0] => [2,3,1,4,5,6] => [6,5,4,1,3,2] => ([(3,4),(3,5)],6) => 9

[1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6) => 10

[1,0,1,1,0,0,1,1,0,1,0,0] => [2,1,4,6,3,5] => [5,3,6,4,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6) => 11

[1,0,1,1,0,0,1,1,1,0,0,0] => [2,1,4,3,5,6] => [6,5,3,4,1,2] => ([(2,5),(3,4)],6) => 9

[1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [3,6,1,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5)],6) => 13

[1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [5,6,3,1,4,2] => ([(0,5),(1,3),(2,4),(2,5)],6) => 11

[1,0,1,1,0,1,1,0,0,1,0,0] => [2,4,1,6,3,5] => [5,3,6,1,4,2] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 12

[1,0,1,1,0,1,1,0,1,0,0,0] => [2,4,6,1,3,5] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 13

[1,0,1,1,0,1,1,1,0,0,0,0] => [2,4,1,3,5,6] => [6,5,3,1,4,2] => ([(2,5),(3,4),(3,5)],6) => 10

[1,0,1,1,1,0,0,0,1,1,0,0] => [2,1,3,5,4,6] => [6,4,5,3,1,2] => ([(2,5),(3,4)],6) => 9

[1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [4,6,3,5,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6) => 11

[1,0,1,1,1,0,0,1,1,0,0,0] => [2,1,5,3,4,6] => [6,4,3,5,1,2] => ([(1,5),(2,5),(3,4)],6) => 10

[1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => [5,6,4,3,1,2] => ([(2,5),(3,4)],6) => 9

[1,0,1,1,1,1,0,0,0,1,0,0] => [2,1,3,6,4,5] => [5,4,6,3,1,2] => ([(1,5),(2,5),(3,4)],6) => 10

[1,0,1,1,1,1,0,0,1,0,0,0] => [2,1,6,3,4,5] => [5,4,3,6,1,2] => ([(0,5),(1,5),(2,5),(3,4)],6) => 11

[1,0,1,1,1,1,1,0,0,0,0,0] => [2,1,3,4,5,6] => [6,5,4,3,1,2] => ([(4,5)],6) => 8

[1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => [2,6,5,4,3,1] => ([(1,2),(1,3),(1,4),(1,5)],6) => 11

[1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,4,5,2,6] => [6,2,5,4,3,1] => ([(2,3),(2,4),(2,5)],6) => 10

[1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,4,2,5,6] => [6,5,2,4,3,1] => ([(3,4),(3,5)],6) => 9

[1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => ([(2,5),(3,4)],6) => 9

[1,1,0,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4,6] => [6,4,2,5,3,1] => ([(2,5),(3,4),(3,5)],6) => 10

[1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => [5,6,4,2,3,1] => ([(2,5),(3,4)],6) => 9

[1,1,0,0,1,1,1,0,0,1,0,0] => [1,3,2,6,4,5] => [5,4,6,2,3,1] => ([(1,5),(2,5),(3,4)],6) => 10

[1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,4,5,6] => [6,5,4,2,3,1] => ([(4,5)],6) => 8

[1,1,0,1,0,0,1,1,0,0,1,0] => [3,1,4,2,6,5] => [5,6,2,4,1,3] => ([(0,5),(1,3),(2,4),(2,5)],6) => 11

[1,1,0,1,0,0,1,1,1,0,0,0] => [3,1,4,2,5,6] => [6,5,2,4,1,3] => ([(2,5),(3,4),(3,5)],6) => 10

[1,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,1,2] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6) => 15

[1,1,0,1,0,1,0,1,1,0,0,0] => [3,4,5,1,2,6] => [6,2,1,5,4,3] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 13

[1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,1,2,5,6] => [6,5,2,1,4,3] => ([(2,4),(2,5),(3,4),(3,5)],6) => 11

[1,1,0,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4,6] => [6,4,5,2,1,3] => ([(1,5),(2,5),(3,4)],6) => 10

[1,1,0,1,1,0,0,1,0,0,1,0] => [3,1,5,2,6,4] => [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 12

[1,1,0,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4,6] => [6,4,2,5,1,3] => ([(1,5),(2,4),(3,4),(3,5)],6) => 11

[1,1,0,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4,6] => [6,4,2,1,5,3] => ([(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 12

>>> Load all 213 entries. <<<

[1,1,0,1,1,1,0,0,0,0,1,0] => [3,1,2,4,6,5] => [5,6,4,2,1,3] => ([(1,5),(2,5),(3,4)],6) => 10

[1,1,0,1,1,1,0,0,0,1,0,0] => [3,1,2,6,4,5] => [5,4,6,2,1,3] => ([(0,5),(1,5),(2,4),(3,4)],6) => 11

[1,1,0,1,1,1,0,0,1,0,0,0] => [3,1,6,2,4,5] => [5,4,2,6,1,3] => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6) => 12

[1,1,0,1,1,1,1,0,0,0,0,0] => [3,1,2,4,5,6] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 9

[1,1,1,0,0,0,1,0,1,0,1,0] => [1,2,4,5,6,3] => [3,6,5,4,2,1] => ([(2,3),(2,4),(2,5)],6) => 10

[1,1,1,0,0,0,1,0,1,1,0,0] => [1,2,4,5,3,6] => [6,3,5,4,2,1] => ([(3,4),(3,5)],6) => 9

[1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => [5,6,3,4,2,1] => ([(2,5),(3,4)],6) => 9

[1,1,1,0,0,0,1,1,0,1,0,0] => [1,2,4,6,3,5] => [5,3,6,4,2,1] => ([(2,5),(3,4),(3,5)],6) => 10

[1,1,1,0,0,0,1,1,1,0,0,0] => [1,2,4,3,5,6] => [6,5,3,4,2,1] => ([(4,5)],6) => 8

[1,1,1,0,0,1,0,0,1,1,0,0] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => ([(2,5),(3,4),(3,5)],6) => 10

[1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,5,6,2,3] => [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 13

[1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,5,2,3,6] => [6,3,2,5,4,1] => ([(2,4),(2,5),(3,4),(3,5)],6) => 11

[1,1,1,0,0,1,1,0,0,0,1,0] => [1,4,2,3,6,5] => [5,6,3,2,4,1] => ([(1,5),(2,5),(3,4)],6) => 10

[1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,2,6,3,5] => [5,3,6,2,4,1] => ([(1,5),(2,4),(3,4),(3,5)],6) => 11

[1,1,1,0,0,1,1,0,1,0,0,0] => [1,4,6,2,3,5] => [5,3,2,6,4,1] => ([(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 12

[1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,2,3,5,6] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 9

[1,1,1,0,1,0,0,1,0,0,1,0] => [4,1,5,2,6,3] => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 13

[1,1,1,0,1,0,0,1,1,0,0,0] => [4,1,5,2,3,6] => [6,3,2,5,1,4] => ([(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 12

[1,1,1,0,1,0,1,0,1,0,0,0] => [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 16

[1,1,1,0,1,0,1,1,0,0,0,0] => [4,5,1,2,3,6] => [6,3,2,1,5,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 13

[1,1,1,0,1,1,0,0,0,0,1,0] => [4,1,2,3,6,5] => [5,6,3,2,1,4] => ([(0,5),(1,5),(2,5),(3,4)],6) => 11

[1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,2,6,3,5] => [5,3,6,2,1,4] => ([(0,5),(1,5),(2,4),(3,4),(3,5)],6) => 12

[1,1,1,0,1,1,0,0,1,0,0,0] => [4,1,6,2,3,5] => [5,3,2,6,1,4] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 13

[1,1,1,0,1,1,0,1,0,0,0,0] => [4,6,1,2,3,5] => [5,3,2,1,6,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 14

[1,1,1,0,1,1,1,0,0,0,0,0] => [4,1,2,3,5,6] => [6,5,3,2,1,4] => ([(2,5),(3,5),(4,5)],6) => 10

[1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => [4,6,5,3,2,1] => ([(3,4),(3,5)],6) => 9

[1,1,1,1,0,0,0,0,1,1,0,0] => [1,2,3,5,4,6] => [6,4,5,3,2,1] => ([(4,5)],6) => 8

[1,1,1,1,0,0,0,1,0,0,1,0] => [1,2,5,3,6,4] => [4,6,3,5,2,1] => ([(2,5),(3,4),(3,5)],6) => 10

[1,1,1,1,0,0,0,1,0,1,0,0] => [1,2,5,6,3,4] => [4,3,6,5,2,1] => ([(2,4),(2,5),(3,4),(3,5)],6) => 11

[1,1,1,1,0,0,0,1,1,0,0,0] => [1,2,5,3,4,6] => [6,4,3,5,2,1] => ([(3,5),(4,5)],6) => 9

[1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,2,6,3,4] => [4,3,6,2,5,1] => ([(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 12

[1,1,1,1,0,0,1,0,1,0,0,0] => [1,5,6,2,3,4] => [4,3,2,6,5,1] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 13

[1,1,1,1,0,0,1,1,0,0,0,0] => [1,5,2,3,4,6] => [6,4,3,2,5,1] => ([(2,5),(3,5),(4,5)],6) => 10

[1,1,1,1,0,1,0,0,1,0,0,0] => [5,1,6,2,3,4] => [4,3,2,6,1,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 14

[1,1,1,1,0,1,0,1,0,0,0,0] => [5,6,1,2,3,4] => [4,3,2,1,6,5] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 15

[1,1,1,1,0,1,1,0,0,0,0,0] => [5,1,2,3,4,6] => [6,4,3,2,1,5] => ([(1,5),(2,5),(3,5),(4,5)],6) => 11

[1,1,1,1,1,0,0,0,0,0,1,0] => [1,2,3,4,6,5] => [5,6,4,3,2,1] => ([(4,5)],6) => 8

[1,1,1,1,1,0,0,0,0,1,0,0] => [1,2,3,6,4,5] => [5,4,6,3,2,1] => ([(3,5),(4,5)],6) => 9

[1,1,1,1,1,0,0,0,1,0,0,0] => [1,2,6,3,4,5] => [5,4,3,6,2,1] => ([(2,5),(3,5),(4,5)],6) => 10

[1,1,1,1,1,0,0,1,0,0,0,0] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => ([(1,5),(2,5),(3,5),(4,5)],6) => 11

[1,1,1,1,1,0,1,0,0,0,0,0] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 12

[1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6) => 7

[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7) => 14

[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [7,1,2,3,4,5,6] => [6,5,4,3,2,1,7] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 14

[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [2,3,4,1,6,7,8,5] => [5,8,7,6,1,4,3,2] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4)],8) => 15

[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [2,3,1,5,6,4,7,8] => [8,7,4,6,5,1,3,2] => ([(2,6),(2,7),(3,4),(3,5)],8) => 13

[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,5,6,1,7,8,4] => [4,8,7,1,6,5,3,2] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7)],8) => 17

[1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,1,4,6,7,5,8] => [8,5,7,6,4,1,3,2] => ([(2,6),(2,7),(3,4),(3,5)],8) => 13

[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,1,4,5,7,8,6] => [6,8,7,5,4,1,3,2] => ([(2,6),(2,7),(3,4),(3,5)],8) => 13

[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5,8,7] => [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8) => 13

[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0] => [2,1,4,3,7,8,5,6] => [6,5,8,7,3,4,1,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8) => 15

[1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [2,1,4,3,5,6,7,8] => [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8) => 11

[1,0,1,1,0,1,0,1,0,0,1,1,1,0,0,0] => [2,4,5,1,6,3,7,8] => [8,7,3,6,1,5,4,2] => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8) => 15

[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [2,4,5,6,7,1,8,3] => [3,8,1,7,6,5,4,2] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7)],8) => 19

[1,0,1,1,0,1,1,0,0,0,1,1,0,1,0,0] => [2,4,1,3,6,8,5,7] => [7,5,8,6,3,1,4,2] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8) => 15

[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0] => [2,4,1,3,7,5,8,6] => [6,8,5,7,3,1,4,2] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8) => 15

[1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => [2,1,3,5,4,6,7,8] => [8,7,6,4,5,3,1,2] => ([(4,7),(5,6)],8) => 11

[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0] => [2,1,5,6,3,4,8,7] => [7,8,4,3,6,5,1,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8) => 15

[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0] => [2,5,1,7,3,4,8,6] => [6,8,4,3,7,1,5,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,4),(3,6)],8) => 17

[1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [2,1,3,4,6,5,7,8] => [8,7,5,6,4,3,1,2] => ([(4,7),(5,6)],8) => 11

[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [2,6,7,1,3,4,8,5] => [5,8,4,3,1,7,6,2] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8) => 19

[1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [2,1,3,4,5,7,6,8] => [8,6,7,5,4,3,1,2] => ([(4,7),(5,6)],8) => 11

[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [2,1,3,4,5,6,8,7] => [7,8,6,5,4,3,1,2] => ([(4,7),(5,6)],8) => 11

[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,6,7,5,8] => [8,5,7,6,2,4,3,1] => ([(2,6),(2,7),(3,4),(3,5)],8) => 13

[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,5,4,6,7,8] => [8,7,6,4,5,2,3,1] => ([(4,7),(5,6)],8) => 11

[1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0] => [1,3,5,6,2,7,4,8] => [8,4,7,2,6,5,3,1] => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8) => 15

[1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,3,2,4,6,5,7,8] => [8,7,5,6,4,2,3,1] => ([(4,7),(5,6)],8) => 11

[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => [1,3,2,4,5,7,6,8] => [8,6,7,5,4,2,3,1] => ([(4,7),(5,6)],8) => 11

[1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => [1,3,2,4,5,6,8,7] => [7,8,6,5,4,2,3,1] => ([(4,7),(5,6)],8) => 11

[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0] => [3,1,4,2,6,8,5,7] => [7,5,8,6,2,4,1,3] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8) => 15

[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,7,5,8,6] => [6,8,5,7,2,4,1,3] => ([(0,7),(1,6),(2,4),(2,6),(3,5),(3,7)],8) => 15

[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [3,4,5,6,7,8,1,2] => [2,1,8,7,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8) => 21

[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0] => [3,4,5,6,1,2,7,8] => [8,7,2,1,6,5,4,3] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) => 17

[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0] => [3,4,1,2,6,5,8,7] => [7,8,5,6,2,1,4,3] => ([(0,6),(0,7),(1,6),(1,7),(2,5),(3,4)],8) => 15

[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0] => [3,4,1,2,7,8,5,6] => [6,5,8,7,2,1,4,3] => ([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(3,6),(3,7)],8) => 17

[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0] => [3,4,1,2,5,6,7,8] => [8,7,6,5,2,1,4,3] => ([(4,6),(4,7),(5,6),(5,7)],8) => 13

[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0] => [3,1,5,6,2,8,4,7] => [7,4,8,2,6,5,1,3] => ([(0,5),(1,4),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8) => 17

[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0] => [3,5,1,7,2,8,4,6] => [6,4,8,2,7,1,5,3] => ([(0,5),(0,7),(1,4),(1,6),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7)],8) => 19

[1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0] => [3,6,7,1,2,8,4,5] => [5,4,8,2,1,7,6,3] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7)],8) => 21

[1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0] => [3,1,2,4,5,8,6,7] => [7,6,8,5,4,2,1,3] => ([(2,7),(3,7),(4,6),(5,6)],8) => 13

[1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,5,3,7,8,6] => [6,8,7,3,5,4,2,1] => ([(2,6),(2,7),(3,4),(3,5)],8) => 13

[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0] => [1,2,4,3,6,5,7,8] => [8,7,5,6,3,4,2,1] => ([(4,7),(5,6)],8) => 11

[1,1,1,0,0,0,1,1,0,1,0,1,0,0,1,0] => [1,2,4,6,7,3,8,5] => [5,8,3,7,6,4,2,1] => ([(2,5),(2,6),(2,7),(3,4),(3,6),(3,7)],8) => 15

[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5,7,6,8] => [8,6,7,5,3,4,2,1] => ([(4,7),(5,6)],8) => 11

[1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0] => [1,2,4,3,5,6,8,7] => [7,8,6,5,3,4,2,1] => ([(4,7),(5,6)],8) => 11

[1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0] => [1,4,5,6,7,2,3,8] => [8,3,2,7,6,5,4,1] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) => 17

[1,1,1,0,0,1,0,1,1,1,1,0,0,0,0,0] => [1,4,5,2,3,6,7,8] => [8,7,6,3,2,5,4,1] => ([(4,6),(4,7),(5,6),(5,7)],8) => 13

[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0] => [1,4,2,3,7,5,6,8] => [8,6,5,7,3,2,4,1] => ([(2,7),(3,7),(4,6),(5,6)],8) => 13

[1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0] => [1,4,2,3,5,8,6,7] => [7,6,8,5,3,2,4,1] => ([(2,7),(3,7),(4,6),(5,6)],8) => 13

[1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0] => [4,1,5,6,8,2,3,7] => [7,3,2,8,6,5,1,4] => ([(0,7),(1,6),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) => 19

[1,1,1,0,1,0,1,1,0,0,1,0,1,0,0,0] => [4,5,1,7,8,2,3,6] => [6,3,2,8,7,1,5,4] => ([(0,6),(0,7),(1,4),(1,5),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) => 21

[1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0] => [4,1,6,2,3,5,7,8] => [8,7,5,3,2,6,1,4] => ([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8) => 15

[1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0] => [4,6,7,1,8,2,3,5] => [5,3,2,8,1,7,6,4] => ([(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) => 23

[1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0] => [4,1,2,3,8,5,6,7] => [7,6,5,8,3,2,1,4] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6)],8) => 15

[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0] => [1,2,3,5,4,7,6,8] => [8,6,7,4,5,3,2,1] => ([(4,7),(5,6)],8) => 11

[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0] => [1,2,3,5,4,6,8,7] => [7,8,6,4,5,3,2,1] => ([(4,7),(5,6)],8) => 11

[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0] => [1,2,5,6,7,8,3,4] => [4,3,8,7,6,5,2,1] => ([(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) => 17

[1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0] => [1,2,5,6,3,4,7,8] => [8,7,4,3,6,5,2,1] => ([(4,6),(4,7),(5,6),(5,7)],8) => 13

[1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4,8,6,7] => [7,6,8,4,3,5,2,1] => ([(2,7),(3,7),(4,6),(5,6)],8) => 13

[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0] => [1,5,2,7,3,4,6,8] => [8,6,4,3,7,2,5,1] => ([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8) => 15

[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8) => 25

[1,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0] => [5,6,1,2,3,4,7,8] => [8,7,4,3,2,1,6,5] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => 17

[1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0] => [5,1,2,8,3,4,6,7] => [7,6,4,3,8,2,1,5] => ([(0,7),(1,7),(2,6),(3,6),(4,6),(4,7),(5,6),(5,7)],8) => 17

[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0] => [1,2,3,4,6,5,8,7] => [7,8,5,6,4,3,2,1] => ([(4,7),(5,6)],8) => 11

[1,1,1,1,1,0,0,0,0,1,0,1,1,0,0,0] => [1,2,3,6,7,4,5,8] => [8,5,4,7,6,3,2,1] => ([(4,6),(4,7),(5,6),(5,7)],8) => 13

[1,1,1,1,1,0,0,0,1,1,0,0,1,0,0,0] => [1,2,6,3,8,4,5,7] => [7,5,4,8,3,6,2,1] => ([(2,7),(3,6),(4,6),(4,7),(5,6),(5,7)],8) => 15

[1,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0] => [1,6,7,2,3,4,5,8] => [8,5,4,3,2,7,6,1] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => 17

[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0] => [6,1,8,2,3,4,5,7] => [7,5,4,3,2,8,1,6] => ([(0,7),(1,6),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => 19

[1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0] => [1,2,3,4,7,8,5,6] => [6,5,8,7,4,3,2,1] => ([(4,6),(4,7),(5,6),(5,7)],8) => 13

[1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0] => [1,2,7,8,3,4,5,6] => [6,5,4,3,8,7,2,1] => ([(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => 17

[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8) => 21

[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8] => [8,7,6,5,4,3,2,1] => ([],8) => 9

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$F_{1} = q^{2}$

$F_{2} = q^{3} + q^{4}$

$F_{3} = q^{4} + 2\ q^{5} + 2\ q^{6}$

$F_{4} = q^{5} + 3\ q^{6} + 5\ q^{7} + 4\ q^{8} + q^{9}$

$F_{5} = q^{6} + 4\ q^{7} + 9\ q^{8} + 12\ q^{9} + 10\ q^{10} + 4\ q^{11} + 2\ q^{12}$

Description

The number of chains of a poset.

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

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

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.