Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤ
Values
{{1}} => [1] => [.,.] => 0
{{1,2}} => [2,1] => [[.,.],.] => 1
{{1},{2}} => [1,2] => [.,[.,.]] => 0
{{1,2,3}} => [2,3,1] => [[.,[.,.]],.] => 2
{{1,2},{3}} => [2,1,3] => [[.,.],[.,.]] => 1
{{1,3},{2}} => [3,2,1] => [[[.,.],.],.] => 2
{{1},{2,3}} => [1,3,2] => [.,[[.,.],.]] => 0
{{1},{2},{3}} => [1,2,3] => [.,[.,[.,.]]] => 0
{{1,2,3,4}} => [2,3,4,1] => [[.,[.,[.,.]]],.] => 3
{{1,2,3},{4}} => [2,3,1,4] => [[.,[.,.]],[.,.]] => 2
{{1,2,4},{3}} => [2,4,3,1] => [[.,[[.,.],.]],.] => 3
{{1,2},{3,4}} => [2,1,4,3] => [[.,.],[[.,.],.]] => 1
{{1,2},{3},{4}} => [2,1,3,4] => [[.,.],[.,[.,.]]] => 1
{{1,3,4},{2}} => [3,2,4,1] => [[[.,.],[.,.]],.] => 3
{{1,3},{2,4}} => [3,4,1,2] => [[.,[.,.]],[.,.]] => 2
{{1,3},{2},{4}} => [3,2,1,4] => [[[.,.],.],[.,.]] => 2
{{1,4},{2,3}} => [4,3,2,1] => [[[[.,.],.],.],.] => 3
{{1},{2,3,4}} => [1,3,4,2] => [.,[[.,[.,.]],.]] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [.,[[.,.],[.,.]]] => 0
{{1,4},{2},{3}} => [4,2,3,1] => [[[.,.],[.,.]],.] => 3
{{1},{2,4},{3}} => [1,4,3,2] => [.,[[[.,.],.],.]] => 0
{{1},{2},{3,4}} => [1,2,4,3] => [.,[.,[[.,.],.]]] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [.,[.,[.,[.,.]]]] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.] => 4
{{1,2,3,4},{5}} => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]] => 3
{{1,2,3,5},{4}} => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.] => 4
{{1,2,3},{4,5}} => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]] => 2
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]] => 2
{{1,2,4,5},{3}} => [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.] => 4
{{1,2,4},{3,5}} => [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]] => 3
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]] => 3
{{1,2,5},{3,4}} => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.] => 4
{{1,2},{3,4,5}} => [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]] => 1
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]] => 1
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.] => 4
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]] => 1
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]] => 1
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]] => 1
{{1,3,4,5},{2}} => [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.] => 4
{{1,3,4},{2,5}} => [3,5,4,1,2] => [[.,[[.,.],.]],[.,.]] => 3
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]] => 3
{{1,3,5},{2,4}} => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.] => 4
{{1,3},{2,4,5}} => [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]] => 2
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]] => 2
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [[[.,.],[[.,.],.]],.] => 4
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]] => 2
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]] => 2
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]] => 2
{{1,4,5},{2,3}} => [4,3,2,5,1] => [[[[.,.],.],[.,.]],.] => 4
{{1,4},{2,3,5}} => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]] => 3
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => 3
{{1,5},{2,3,4}} => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.] => 4
{{1},{2,3,4,5}} => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]] => 0
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]] => 0
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.] => 4
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [.,[[.,[[.,.],.]],.]] => 0
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]] => 0
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]] => 0
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.] => 4
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]] => 3
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]] => 3
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]] => 3
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => 4
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [.,[[[.,.],[.,.]],.]] => 0
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]] => 0
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]] => 0
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.] => 4
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]] => 0
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]] => 0
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]] => 0
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.] => 4
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]] => 0
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]] => 0
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]] => 0
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]] => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [[.,[.,[.,[.,[.,.]]]]],.] => 5
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [[.,[.,[.,[.,.]]]],[.,.]] => 4
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [[.,[.,[.,[[.,.],.]]]],.] => 5
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [[.,[.,[.,.]]],[[.,.],.]] => 3
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [[.,[.,[.,.]]],[.,[.,.]]] => 3
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [[.,[.,[[.,.],[.,.]]]],.] => 5
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [[.,[.,[.,[.,.]]]],[.,.]] => 4
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [[.,[.,[[.,.],.]]],[.,.]] => 4
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [[.,[.,[[[.,.],.],.]]],.] => 5
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [[.,[.,.]],[[.,[.,.]],.]] => 2
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [[.,[.,.]],[[.,.],[.,.]]] => 2
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [[.,[.,[[.,.],[.,.]]]],.] => 5
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [[.,[.,.]],[[[.,.],.],.]] => 2
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [[.,[.,.]],[.,[[.,.],.]]] => 2
{{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] => [[.,[[.,.],[.,[.,.]]]],.] => 5
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [[.,[.,[[.,.],.]]],[.,.]] => 4
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [[.,[[.,.],[.,.]]],[.,.]] => 4
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [[.,[[.,[.,[.,.]]],.]],.] => 5
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [[.,[.,[.,.]]],[[.,.],.]] => 3
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [[.,[.,[.,.]]],[.,[.,.]]] => 3
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [[.,[[.,.],[[.,.],.]]],.] => 5
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [[.,[.,[.,.]]],[[.,.],.]] => 3
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [[.,[[.,.],.]],[[.,.],.]] => 3
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [[.,[[.,.],.]],[.,[.,.]]] => 3
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [[.,[[[.,.],.],[.,.]]],.] => 5
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Description
The size of the left subtree of a binary tree.
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.
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.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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