Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000501: Permutations ⟶ ℤ
Values
[.,.] => [1] => [1] => 1
[.,[.,.]] => [2,1] => [1,2] => 1
[[.,.],.] => [1,2] => [2,1] => 2
[.,[.,[.,.]]] => [3,2,1] => [1,3,2] => 1
[.,[[.,.],.]] => [2,3,1] => [1,2,3] => 1
[[.,.],[.,.]] => [1,3,2] => [2,1,3] => 2
[[.,[.,.]],.] => [2,1,3] => [3,2,1] => 3
[[[.,.],.],.] => [1,2,3] => [2,3,1] => 3
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,4,3,2] => 1
[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,4,2,3] => 1
[.,[[.,.],[.,.]]] => [2,4,3,1] => [1,2,4,3] => 1
[.,[[.,[.,.]],.]] => [3,2,4,1] => [1,3,2,4] => 1
[.,[[[.,.],.],.]] => [2,3,4,1] => [1,2,3,4] => 1
[[.,.],[.,[.,.]]] => [1,4,3,2] => [2,1,4,3] => 2
[[.,.],[[.,.],.]] => [1,3,4,2] => [2,1,3,4] => 2
[[.,[.,.]],[.,.]] => [2,1,4,3] => [3,2,1,4] => 3
[[[.,.],.],[.,.]] => [1,2,4,3] => [2,3,1,4] => 3
[[.,[.,[.,.]]],.] => [3,2,1,4] => [4,3,2,1] => 4
[[.,[[.,.],.]],.] => [2,3,1,4] => [4,2,3,1] => 4
[[[.,.],[.,.]],.] => [1,3,2,4] => [2,4,3,1] => 4
[[[.,[.,.]],.],.] => [2,1,3,4] => [3,2,4,1] => 4
[[[[.,.],.],.],.] => [1,2,3,4] => [2,3,4,1] => 4
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,5,4,3,2] => 1
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,5,4,2,3] => 1
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [1,5,2,4,3] => 1
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [1,5,3,2,4] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,5,2,3,4] => 1
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [1,2,5,4,3] => 1
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [1,2,5,3,4] => 1
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [1,3,2,5,4] => 1
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [1,2,3,5,4] => 1
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [1,4,3,2,5] => 1
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,4,2,3,5] => 1
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [1,2,4,3,5] => 1
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [1,3,2,4,5] => 1
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,2,3,4,5] => 1
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [2,1,5,4,3] => 2
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [2,1,5,3,4] => 2
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [2,1,3,5,4] => 2
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [2,1,4,3,5] => 2
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [2,1,3,4,5] => 2
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [3,2,1,5,4] => 3
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [3,2,1,4,5] => 3
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [2,3,1,5,4] => 3
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [2,3,1,4,5] => 3
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [4,3,2,1,5] => 4
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [4,2,3,1,5] => 4
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [2,4,3,1,5] => 4
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [3,2,4,1,5] => 4
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [2,3,4,1,5] => 4
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [5,4,3,2,1] => 5
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [5,4,2,3,1] => 5
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [5,2,4,3,1] => 5
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [5,3,2,4,1] => 5
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [5,2,3,4,1] => 5
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [2,5,4,3,1] => 5
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [2,5,3,4,1] => 5
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [3,2,5,4,1] => 5
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [2,3,5,4,1] => 5
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [4,3,2,5,1] => 5
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [4,2,3,5,1] => 5
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [2,4,3,5,1] => 5
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [3,2,4,5,1] => 5
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [2,3,4,5,1] => 5
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,6,5,4,3,2] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [1,6,5,4,2,3] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [1,6,5,2,4,3] => 1
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [1,6,5,3,2,4] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [1,6,5,2,3,4] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [1,6,2,5,4,3] => 1
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [1,6,2,5,3,4] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [1,6,3,2,5,4] => 1
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [1,6,2,3,5,4] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [1,6,4,3,2,5] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [1,6,4,2,3,5] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [1,6,2,4,3,5] => 1
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [1,6,3,2,4,5] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [1,6,2,3,4,5] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [1,2,6,5,4,3] => 1
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [1,2,6,5,3,4] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [1,2,6,3,5,4] => 1
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [1,2,6,4,3,5] => 1
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [1,2,6,3,4,5] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [1,3,2,6,5,4] => 1
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [1,3,2,6,4,5] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [1,2,3,6,5,4] => 1
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [1,2,3,6,4,5] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [1,4,3,2,6,5] => 1
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [1,4,2,3,6,5] => 1
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [1,2,4,3,6,5] => 1
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [1,3,2,4,6,5] => 1
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [1,2,3,4,6,5] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [1,5,4,3,2,6] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [1,5,4,2,3,6] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [1,5,2,4,3,6] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [1,5,3,2,4,6] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [1,5,2,3,4,6] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [1,2,5,4,3,6] => 1
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [1,2,5,3,4,6] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [1,3,2,5,4,6] => 1
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [1,2,3,5,4,6] => 1
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Description
The size of the first part in the decomposition of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is defined to be the smallest $k > 0$ such that $\{\pi(1),\ldots,\pi(k)\} = \{1,\ldots,k\}$. This statistic is undefined for the empty permutation.
For the number of parts in the decomposition see St000056The decomposition (or block) number of a permutation..
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is defined to be the smallest $k > 0$ such that $\{\pi(1),\ldots,\pi(k)\} = \{1,\ldots,k\}$. This statistic is undefined for the empty permutation.
For the number of parts in the decomposition see St000056The decomposition (or block) number of a permutation..
Map
Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $\pi^{-1}c$ where $c = (1,\ldots,n)$ is the long cycle.
Map
to 312-avoiding permutation
Description
Return a 312-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 minimal element of this Sylvester class.
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 minimal element of this Sylvester class.
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