- FindStat

Identifier

Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000030: Permutations ⟶ ℤ

Values

[1,0] => [.,.] => [1] => [1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 196 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

click to show known generating functions

$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + q + 3\ q^{2}$

$F_{4} = 1 + q + 3\ q^{2} + 7\ q^{3} + 2\ q^{4}$

$F_{5} = 1 + q + 3\ q^{2} + 7\ q^{3} + 17\ q^{4} + 8\ q^{5} + 5\ q^{6}$

$F_{6} = 1 + q + 3\ q^{2} + 7\ q^{3} + 17\ q^{4} + 39\ q^{5} + 29\ q^{6} + 21\ q^{7} + 11\ q^{8} + 3\ q^{9}$

Description

The sum of the descent differences of a permutations.
This statistic is given by

$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} (\pi_i-\pi_{i+1}).$
See St000111The sum of the descent tops (or Genocchi descents) of a permutation. and St000154The sum of the descent bottoms of a permutation. for the sum of the descent tops and the descent bottoms, respectively. This statistic was studied in [1] and [2] where is was called the drop of a permutation.

Map

inverse

Description

Sends a permutation to its inverse.

Map

logarithmic height to pruning number

Description

Francon's map from Dyck paths to binary trees.
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].

Map

to 132-avoiding permutation

Description

Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.