- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000118: Binary trees ⟶ ℤ

Values

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

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

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

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

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

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

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

{{1},{2},{3}} => [1,2,3] => [.,[.,[.,.]]] => 1

{{1,2,3,4}} => [2,3,4,1] => [[.,[.,[.,.]]],.] => 1

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

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

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

{{1,2},{3},{4}} => [2,1,3,4] => [[.,.],[.,[.,.]]] => 1

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

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

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

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

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

{{1},{2,3},{4}} => [1,3,2,4] => [.,[[.,.],[.,.]]] => 1

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

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

{{1},{2},{3,4}} => [1,2,4,3] => [.,[.,[[.,.],.]]] => 1

{{1},{2},{3},{4}} => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 2

{{1,2,3,4,5}} => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => 2

{{1,2,3,4},{5}} => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => 1

{{1,2,3,5},{4}} => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.] => 1

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

{{1,2,3},{4},{5}} => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]] => 1

{{1,2,4,5},{3}} => [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.] => 1

{{1,2,4},{3,5}} => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]] => 1

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

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

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

{{1,2},{3,4},{5}} => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => 1

{{1,2,5},{3},{4}} => [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.] => 1

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

{{1,2},{3},{4,5}} => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => 1

{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => 2

{{1,3,4,5},{2}} => [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.] => 1

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

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

{{1,3,5},{2,4}} => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.] => 1

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

{{1,3},{2,4},{5}} => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => 1

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

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

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

{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => 1

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

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

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

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

{{1},{2,3,4,5}} => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => 1

{{1},{2,3,4},{5}} => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => 1

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

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

{{1},{2,3},{4,5}} => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => 1

{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 2

{{1,4,5},{2},{3}} => [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.] => 1

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

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

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

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

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

{{1},{2,4},{3,5}} => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 1

{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => 1

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

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

{{1},{2},{3,4,5}} => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => 1

{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 2

{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.] => 1

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

{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => 1

{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 2

{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1155 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 10 + 4\ q + q^{2}$

$F_{5} = 25 + 21\ q + 5\ q^{2} + q^{3}$

$F_{6} = 71 + 88\ q + 37\ q^{2} + 6\ q^{3} + q^{4}$

$F_{7} = 217 + 370\ q + 221\ q^{2} + 61\ q^{3} + 7\ q^{4} + q^{5}$

Description

The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree.
oeis:A001006 counts binary trees avoiding this pattern.

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length

$0$ to the empty tree, and sending a permutation

$\sigma$ of length

$n \geq 1$ to a root node with two subtrees

$L$ and

$R$ by splitting

$\sigma$ at the index

$\sigma^{-1}(1)$ , normalizing both sides again to permutations and sending the permutations on the left and on the right of

$\sigma^{-1}(1)$ to the trees

$L$ and

$R$ , respectively.