- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,2,1] => [1,0,1,0,1,0,1,0] => [4,3,2,1] => ([],4) => 24

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

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

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

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

[5,2] => [1,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6) => 18

[5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6) => 18

[4,3] => [1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 20

[4,2,1] => [1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5) => 30

[4,1,1,1] => [1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 20

[3,3,1] => [1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5) => 20

[3,2,2] => [1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5) => 20

[3,2,1,1] => [1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5) => 30

[3,1,1,1,1] => [1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6) => 18

[2,2,2,1] => [1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => ([(2,3),(3,4)],5) => 20

[2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => ([(1,5),(2,3),(3,5),(5,4)],6) => 18

[5,3] => [1,1,1,1,0,0,0,1,0,0,1,0] => [6,4,1,2,3,5] => ([(1,5),(2,3),(3,4),(4,5)],6) => 24

[5,2,1] => [1,1,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1,4,5] => ([(1,5),(2,5),(3,5),(5,4)],6) => 36

[5,1,1,1] => [1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => ([(1,5),(2,3),(3,4),(4,5)],6) => 24

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

[4,3,1] => [1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 40

[4,2,2] => [1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => ([(1,4),(2,3)],5) => 30

[4,2,1,1] => [1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 40

[4,1,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6) => 24

[3,3,2] => [1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => ([(1,4),(2,3)],5) => 30

[3,3,1,1] => [1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => ([(1,4),(2,3)],5) => 30

[3,2,2,1] => [1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => ([(2,4),(3,4)],5) => 40

[3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => ([(1,5),(2,5),(3,5),(5,4)],6) => 36

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

[2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => ([(1,5),(2,3),(3,4),(4,5)],6) => 24

[5,4] => [1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6) => 30

[5,3,1] => [1,1,1,0,1,0,0,1,0,0,1,0] => [6,4,2,1,3,5] => ([(1,5),(2,4),(3,4),(4,5)],6) => 48

[5,2,2] => [1,1,1,0,0,1,1,0,0,0,1,0] => [6,3,4,1,2,5] => ([(1,4),(2,3),(3,5),(4,5)],6) => 36

[5,2,1,1] => [1,1,0,1,1,0,1,0,0,0,1,0] => [6,3,2,4,1,5] => ([(1,5),(2,4),(3,4),(4,5)],6) => 48

[5,1,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6) => 30

[4,4,1] => [1,1,1,0,1,0,0,0,1,1,0,0] => [5,6,2,1,3,4] => ([(0,5),(1,5),(2,3),(5,4)],6) => 30

[4,3,2] => [1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => ([(3,4)],5) => 60

[4,3,1,1] => [1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => ([(3,4)],5) => 60

[4,2,2,1] => [1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => ([(3,4)],5) => 60

[4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => ([(1,5),(2,4),(3,4),(4,5)],6) => 48

[3,3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6) => 20

[3,3,2,1] => [1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => ([(3,4)],5) => 60

[3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => ([(1,4),(2,3),(3,5),(4,5)],6) => 36

[3,2,2,2] => [1,1,0,0,1,1,1,0,1,0,0,0] => [4,3,5,6,1,2] => ([(0,5),(1,5),(2,3),(5,4)],6) => 30

[3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => ([(1,5),(2,4),(3,4),(4,5)],6) => 48

[2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => ([(2,3),(3,5),(5,4)],6) => 30

[5,4,1] => [1,1,1,0,1,0,0,0,1,0,1,0] => [6,5,2,1,3,4] => ([(2,5),(3,5),(5,4)],6) => 60

[5,3,2] => [1,1,1,0,0,1,0,1,0,0,1,0] => [6,4,3,1,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6) => 72

[5,3,1,1] => [1,1,0,1,1,0,0,1,0,0,1,0] => [6,4,2,3,1,5] => ([(1,5),(2,5),(3,4),(4,5)],6) => 72

[5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0] => [6,3,4,2,1,5] => ([(1,5),(2,5),(3,4),(4,5)],6) => 72

[5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => ([(2,5),(3,5),(5,4)],6) => 60

[4,4,2] => [1,1,1,0,0,1,0,0,1,1,0,0] => [5,6,3,1,2,4] => ([(0,5),(1,3),(2,4),(4,5)],6) => 45

[4,4,1,1] => [1,1,0,1,1,0,0,0,1,1,0,0] => [5,6,2,3,1,4] => ([(0,5),(1,3),(2,4),(4,5)],6) => 45

[4,3,3] => [1,1,1,0,0,0,1,1,0,1,0,0] => [5,4,6,1,2,3] => ([(0,5),(1,5),(2,3),(3,4)],6) => 40

[4,3,2,1] => [1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => ([],5) => 120

[4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => 72

[4,2,2,2] => [1,1,0,0,1,1,1,0,0,1,0,0] => [5,3,4,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6) => 45

[4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => 72

[3,3,3,1] => [1,1,0,1,0,0,1,1,1,0,0,0] => [4,5,6,2,1,3] => ([(0,5),(1,5),(2,3),(3,4)],6) => 40

[3,3,2,2] => [1,1,0,0,1,1,0,1,1,0,0,0] => [4,5,3,6,1,2] => ([(0,5),(1,3),(2,4),(4,5)],6) => 45

[3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => ([(1,5),(2,5),(3,4),(4,5)],6) => 72

[3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => ([(2,5),(3,5),(5,4)],6) => 60

[5,4,2] => [1,1,1,0,0,1,0,0,1,0,1,0] => [6,5,3,1,2,4] => ([(2,5),(3,4),(4,5)],6) => 90

[5,4,1,1] => [1,1,0,1,1,0,0,0,1,0,1,0] => [6,5,2,3,1,4] => ([(2,5),(3,4),(4,5)],6) => 90

[5,3,3] => [1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6) => 60

[5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0] => [6,4,3,2,1,5] => ([(1,5),(2,5),(3,5),(4,5)],6) => 144

[5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => ([(2,5),(3,4),(4,5)],6) => 90

[5,2,2,2] => [1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => ([(1,3),(2,4),(4,5)],6) => 60

[5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => ([(2,5),(3,4),(4,5)],6) => 90

[4,4,3] => [1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => ([(1,3),(2,4),(4,5)],6) => 60

[4,4,2,1] => [1,1,0,1,0,1,0,0,1,1,0,0] => [5,6,3,2,1,4] => ([(0,5),(1,5),(2,5),(3,4)],6) => 90

[4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => ([(1,3),(2,4),(4,5)],6) => 60

[4,3,3,1] => [1,1,0,1,0,0,1,1,0,1,0,0] => [5,4,6,2,1,3] => ([(0,5),(1,5),(2,4),(3,4)],6) => 80

[4,3,2,2] => [1,1,0,0,1,1,0,1,0,1,0,0] => [5,4,3,6,1,2] => ([(0,5),(1,5),(2,5),(3,4)],6) => 90

[4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => ([(1,5),(2,5),(3,5),(4,5)],6) => 144

[4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => ([(2,5),(3,4),(4,5)],6) => 90

[3,3,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => ([(1,3),(2,4),(4,5)],6) => 60

[3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => ([(1,3),(2,4),(4,5)],6) => 60

[3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => ([(2,5),(3,4),(4,5)],6) => 90

>>> Load all 142 entries. <<<

[5,4,3] => [1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => ([(3,4),(4,5)],6) => 120

[5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0] => [6,5,3,2,1,4] => ([(2,5),(3,5),(4,5)],6) => 180

[5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => ([(3,4),(4,5)],6) => 120

[5,3,3,1] => [1,1,0,1,0,0,1,1,0,0,1,0] => [6,4,5,2,1,3] => ([(1,5),(2,5),(3,4)],6) => 120

[5,3,2,2] => [1,1,0,0,1,1,0,1,0,0,1,0] => [6,4,3,5,1,2] => ([(1,5),(2,5),(3,4)],6) => 120

[5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => ([(2,5),(3,5),(4,5)],6) => 180

[5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => ([(3,4),(4,5)],6) => 120

[4,4,3,1] => [1,1,0,1,0,0,1,0,1,1,0,0] => [5,6,4,2,1,3] => ([(1,5),(2,5),(3,4)],6) => 120

[4,4,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6) => 90

[4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => ([(1,5),(2,5),(3,4)],6) => 120

[4,3,3,2] => [1,1,0,0,1,0,1,1,0,1,0,0] => [5,4,6,3,1,2] => ([(1,5),(2,5),(3,4)],6) => 120

[4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => ([(1,5),(2,5),(3,4)],6) => 120

[4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => ([(2,5),(3,5),(4,5)],6) => 180

[3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => ([(3,4),(4,5)],6) => 120

[5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,0] => [6,5,4,2,1,3] => ([(3,5),(4,5)],6) => 240

[5,4,2,2] => [1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => ([(2,5),(3,4)],6) => 180

[5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => ([(3,5),(4,5)],6) => 240

[5,3,3,2] => [1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => ([(2,5),(3,4)],6) => 180

[5,3,3,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => ([(2,5),(3,4)],6) => 180

[5,3,2,2,1] => [1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => ([(3,5),(4,5)],6) => 240

[4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => ([(2,5),(3,4)],6) => 180

[4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => ([(2,5),(3,4)],6) => 180

[4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => ([(2,5),(3,4)],6) => 180

[4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => ([(3,5),(4,5)],6) => 240

[5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => ([(4,5)],6) => 360

[5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => ([(4,5)],6) => 360

[5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => ([(4,5)],6) => 360

[5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => ([(4,5)],6) => 360

[4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => ([(4,5)],6) => 360

[5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => ([],6) => 720

[4,4,4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8) => 70

[7,6,5,4,3,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,7,6,5,4,3,2,1] => ([],8) => 40320

[6,6,4,4,3,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [7,8,5,6,4,3,2,1] => ([(4,7),(5,6)],8) => 10080

[6,6,5,3,3,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [7,8,6,4,5,3,2,1] => ([(4,7),(5,6)],8) => 10080

[6,6,5,4,2,2,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [7,8,6,5,3,4,2,1] => ([(4,7),(5,6)],8) => 10080

[6,6,5,4,3,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [7,8,6,5,4,2,3,1] => ([(4,7),(5,6)],8) => 10080

[6,5,5,4,2,1,1] => [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [7,6,8,5,3,2,4,1] => ([(2,7),(3,7),(4,6),(5,6)],8) => 4480

[6,6,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,8,6,5,4,3,1,2] => ([(4,7),(5,6)],8) => 10080

[7,5,5,4,3,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => [8,6,7,5,4,3,1,2] => ([(4,7),(5,6)],8) => 10080

[7,6,4,4,3,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => [8,7,5,6,4,3,1,2] => ([(4,7),(5,6)],8) => 10080

[6,5,5,4,3,1] => [1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0] => [7,6,8,5,4,2,1,3] => ([(2,7),(3,7),(4,6),(5,6)],8) => 4480

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

$F_{2} = 2\ q^{3}$

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

$F_{4} = 2\ q^{5} + q^{6} + 2\ q^{8}$

$F_{5} = 2\ q^{6} + 2\ q^{10} + 3\ q^{12}$

Description

The number of linear extensions of a poset.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.

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

$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$