Identifier
-
Mp00014:
Binary trees
—to 132-avoiding permutation⟶
Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001085: Permutations ⟶ ℤ
Values
[.,.] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [2,1] => 0
[[.,.],.] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]] => [2,3,1] => [2,1,3] => 1
[[.,.],[.,.]] => [3,1,2] => [3,1,2] => 0
[[.,[.,.]],.] => [2,1,3] => [1,3,2] => 0
[[[.,.],.],.] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,1,2] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [3,2,4,1] => 1
[.,[[.,.],[.,.]]] => [4,2,3,1] => [4,3,2,1] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [2,1,3,4] => 1
[.,[[[.,.],.],.]] => [2,3,4,1] => [2,1,4,3] => 1
[[.,.],[.,[.,.]]] => [4,3,1,2] => [4,2,1,3] => 0
[[.,.],[[.,.],.]] => [3,4,1,2] => [3,1,4,2] => 1
[[.,[.,.]],[.,.]] => [4,2,1,3] => [4,2,3,1] => 0
[[[.,.],.],[.,.]] => [4,1,2,3] => [4,1,3,2] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,4,3,2] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,4,2,3] => 0
[[[.,.],[.,.]],.] => [3,1,2,4] => [1,3,4,2] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => [1,3,2,4] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,1,2,3] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [4,3,5,1,2] => 1
[.,[.,[[.,.],[.,.]]]] => [5,3,4,2,1] => [5,4,2,1,3] => 0
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [3,2,4,5,1] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [3,2,5,4,1] => 1
[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => [5,4,1,3,2] => 0
[.,[[.,.],[[.,.],.]]] => [4,5,2,3,1] => [4,3,5,2,1] => 1
[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => [5,4,3,1,2] => 0
[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => [5,4,3,2,1] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [2,1,3,4,5] => 1
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [2,1,4,3,5] => 1
[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => [2,1,3,5,4] => 1
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [2,1,5,3,4] => 1
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [2,1,5,4,3] => 1
[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => [5,3,1,2,4] => 0
[[.,.],[.,[[.,.],.]]] => [4,5,3,1,2] => [4,2,5,1,3] => 1
[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => [5,3,2,1,4] => 0
[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => [3,1,4,5,2] => 1
[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => [3,1,5,4,2] => 1
[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => [5,3,1,4,2] => 0
[[.,[.,.]],[[.,.],.]] => [4,5,2,1,3] => [4,2,5,3,1] => 1
[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => [5,2,1,4,3] => 0
[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => [4,1,5,3,2] => 1
[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => [5,3,4,2,1] => 0
[[.,[[.,.],.]],[.,.]] => [5,2,3,1,4] => [5,3,4,1,2] => 0
[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => [5,2,4,3,1] => 0
[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => [5,2,4,1,3] => 0
[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => [5,1,4,2,3] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,5,4,3,2] => 0
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,5,4,2,3] => 0
[[.,[[.,.],[.,.]]],.] => [4,2,3,1,5] => [1,5,3,4,2] => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,5,3,2,4] => 0
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,5,2,3,4] => 0
[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => [1,4,5,3,2] => 0
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => [1,4,5,2,3] => 0
[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => [1,4,3,5,2] => 0
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => [1,3,4,5,2] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,4,3,2,5] => 0
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,4,2,3,5] => 0
[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => [1,3,4,2,5] => 0
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,3,2,4,5] => 0
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,5,1,2,3,4] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [5,4,6,1,2,3] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [6,5,2,1,3,4] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [4,3,5,6,1,2] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [4,3,6,5,1,2] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [6,5,1,3,2,4] => 0
[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [5,4,6,2,1,3] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [6,5,3,1,2,4] => 0
[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [6,5,3,2,1,4] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [3,2,4,5,6,1] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [3,2,5,4,6,1] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [3,2,4,6,5,1] => 1
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [3,2,6,4,5,1] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [3,2,6,5,4,1] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [6,5,1,2,4,3] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => [5,4,6,1,3,2] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [6,5,2,1,4,3] => 0
[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [4,3,5,6,2,1] => 1
[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [4,3,6,5,2,1] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [6,5,1,4,2,3] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => [5,4,6,3,1,2] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [6,5,1,4,3,2] => 0
[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [5,4,6,3,2,1] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [6,5,4,1,2,3] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [6,5,4,2,1,3] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [6,5,4,1,3,2] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [6,5,4,3,1,2] => 0
[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [6,5,4,3,2,1] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [2,1,3,4,5,6] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [2,1,4,3,5,6] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [2,1,3,5,4,6] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [2,1,5,3,4,6] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [2,1,5,4,3,6] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [2,1,3,4,6,5] => 1
[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [2,1,4,3,6,5] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [2,1,3,6,4,5] => 1
[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [2,1,3,6,5,4] => 1
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Description
The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Map
ones to leading
Description
The unique permutation obtained by applying the Foata-Riordan map to obtain a Prüfer code, then prepending zero and cyclically shifting.
Let $c_1,\dots, c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let $c_0 = 0$.
This map returns the a unique permutation $q_1,\dots, q_n$ such that $q_i - c_{i-1}$ is constant modulo $n+1$.
This map is Mp00299ones to leading restricted to permutations.
Let $c_1,\dots, c_{n-1}$ be the Prüfer code obtained via the Foata-Riordan map described in [1, eq (1.2)] and let $c_0 = 0$.
This map returns the a unique permutation $q_1,\dots, q_n$ such that $q_i - c_{i-1}$ is constant modulo $n+1$.
This map is Mp00299ones to leading restricted to permutations.
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.
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.
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