- FindStat

Identifier

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000100: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [.,[[.,.],[.,[[.,.],[.,.]]]]] => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7) => 10

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

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

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

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

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

[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [.,[[[.,.],.],[.,[.,[.,.]]]]] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7) => 10

[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [.,[[[.,.],.],[.,[[.,.],.]]]] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7) => 10

[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [.,[[[.,.],.],[[.,.],[.,.]]]] => ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7) => 20

[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [.,[[[.,.],.],[[[.,.],.],.]]] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7) => 10

[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [.,[[[[.,.],.],.],[.,[.,.]]]] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7) => 10

[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [.,[[[[.,.],.],.],[[.,.],.]]] => ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7) => 10

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

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

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

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

[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[.,.],[.,[.,[[.,.],[.,.]]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7) => 12

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

[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[.,.],[.,[[.,.],[.,[.,.]]]]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => 18

[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[.,.],[.,[[.,.],[[.,.],.]]]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => 18

[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => [[.,.],[.,[[[.,.],.],[.,.]]]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => 18

>>> Load all 153 entries. <<<

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

[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[.,.],[[.,.],[.,[.,[.,.]]]]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => 24

[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[.,.],[[.,.],[.,[[.,.],.]]]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => 24

[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[.,.],[[.,.],[[.,.],[.,.]]]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => 48

[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => [[.,.],[[.,.],[[[.,.],.],.]]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => 24

[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => [[.,.],[[[.,.],.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => 36

[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => [[.,.],[[[.,.],.],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => 36

[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => [[.,.],[[[[.,.],.],.],[.,.]]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => 24

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

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

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

[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => [[[.,.],.],[.,[[.,.],[.,.]]]] => ([(0,5),(1,5),(2,3),(3,6),(4,6),(5,4)],7) => 30

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

[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => [[[.,.],.],[[.,.],[.,[.,.]]]] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7) => 45

[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => [[[.,.],.],[[.,.],[[.,.],.]]] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7) => 45

[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => [[[.,.],.],[[[.,.],.],[.,.]]] => ([(0,5),(1,3),(2,4),(3,6),(4,5),(5,6)],7) => 45

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

[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => [[[[.,.],.],.],[.,[.,[.,.]]]] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7) => 20

[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => [[[[.,.],.],.],[.,[[.,.],.]]] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7) => 20

[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => [[[[.,.],.],.],[[.,.],[.,.]]] => ([(0,5),(1,5),(2,3),(3,4),(4,6),(5,6)],7) => 40

[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => [[[[.,.],.],.],[[[.,.],.],.]] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7) => 20

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

[1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

[1,1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [.,[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

[1,1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [.,[.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

[1,1,1,1,1,5] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [.,[.,[.,[.,[.,[[[[[.,.],.],.],.],.]]]]]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

[1,1,1,1,6] => [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [.,[.,[.,[.,[[[[[[.,.],.],.],.],.],.]]]]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

[1,1,1,7] => [1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [.,[.,[.,[[[[[[[.,.],.],.],.],.],.],.]]]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

[1,1,8] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [.,[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1

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$F_{2} = 2\ q$

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

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

$F_{5} = 5\ q + q^{2} + 3\ q^{3} + 4\ q^{4} + 2\ q^{6} + q^{8}$

$F_{6} = 6\ q + q^{2} + 3\ q^{3} + 4\ q^{4} + 5\ q^{5} + 2\ q^{6} + q^{8} + 6\ q^{10} + 3\ q^{15} + q^{20}$

$F_{7} = 7\ q + q^{2} + 3\ q^{3} + 4\ q^{4} + 5\ q^{5} + 8\ q^{6} + q^{8} + 6\ q^{10} + q^{12} + 9\ q^{15} + 3\ q^{18} + 4\ q^{20} + 4\ q^{24} + q^{30} + 2\ q^{36} + q^{40} + 3\ q^{45} + q^{48}$

Description

The number of linear extensions of a poset.

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

to binary tree: up step, left tree, down step, right tree

Description

Return the binary tree corresponding to the Dyck path under the transformation up step - left tree - down step - right tree.
A Dyck path

$D$ of semilength

$n$ with

$n > 1$ may be uniquely decomposed into

$1L0R$ for Dyck paths L,R of respective semilengths

$n_1, n_2$ with

$n_1 + n_2 = n-1$ .
This map sends

$D$ to the binary tree

$T$ consisting of a root node with a left child according to

$L$ and a right child according to

$R$ and then recursively proceeds.
The base case of the unique Dyck path of semilength

$1$ is sent to a single node.

Map

to poset

Description

Return the poset obtained by interpreting the tree as a Hasse diagram.