Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000779: Permutations ⟶ ℤ
Values
[.,[.,.]] => [2,1] => [2,1] => [2,1] => 0
[[.,.],.] => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]] => [2,3,1] => [2,1,3] => [2,1,3] => 0
[[.,.],[.,.]] => [1,3,2] => [2,3,1] => [3,1,2] => 0
[[.,[.,.]],.] => [2,1,3] => [1,3,2] => [1,3,2] => 0
[[[.,.],.],.] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,1,2] => [3,4,2,1] => 1
[.,[.,[[.,.],.]]] => [3,4,2,1] => [3,2,4,1] => [4,2,1,3] => 0
[.,[[.,.],[.,.]]] => [2,4,3,1] => [3,2,1,4] => [3,2,1,4] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[[.,.],[.,[.,.]]] => [1,4,3,2] => [3,4,2,1] => [4,3,1,2] => 0
[[.,.],[[.,.],.]] => [1,3,4,2] => [2,3,1,4] => [3,1,2,4] => 0
[[.,[.,.]],[.,.]] => [2,1,4,3] => [2,4,3,1] => [4,1,3,2] => 0
[[[.,.],.],[.,.]] => [1,2,4,3] => [2,3,4,1] => [4,1,2,3] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,4,3,2] => [1,4,3,2] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 1
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,1,2,3] => [3,4,5,2,1] => 1
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [4,3,5,1,2] => [4,5,2,1,3] => 1
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [4,3,1,5,2] => [3,5,2,1,4] => 1
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [3,2,4,5,1] => [5,2,1,3,4] => 0
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [3,2,5,4,1] => [5,2,1,4,3] => 0
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [4,3,1,2,5] => [3,4,2,1,5] => 1
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [3,2,5,1,4] => [4,2,1,5,3] => 1
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [3,2,1,5,4] => [3,2,1,5,4] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [2,1,4,5,3] => [2,1,5,3,4] => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [2,1,5,3,4] => [2,1,4,5,3] => 1
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [2,1,5,4,3] => [2,1,5,4,3] => 0
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [4,5,3,1,2] => [4,5,3,1,2] => 1
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [3,4,2,5,1] => [5,3,1,2,4] => 0
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [3,4,2,1,5] => [4,3,1,2,5] => 0
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [2,3,1,4,5] => [3,1,2,4,5] => 0
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [2,3,1,5,4] => [3,1,2,5,4] => 0
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [3,5,4,2,1] => [5,4,1,3,2] => 0
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [2,4,3,1,5] => [4,1,3,2,5] => 0
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => 0
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [2,3,4,1,5] => [4,1,2,3,5] => 0
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [2,5,4,3,1] => [5,1,4,3,2] => 0
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [2,5,3,4,1] => [5,1,3,4,2] => 1
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [2,3,5,4,1] => [5,1,2,4,3] => 0
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [2,4,3,5,1] => [5,1,3,2,4] => 0
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [2,3,4,5,1] => [5,1,2,3,4] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,5,4,2,3] => [1,4,5,3,2] => 1
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [1,5,2,4,3] => [1,3,5,4,2] => 1
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,5,3,2,4] => [1,4,3,5,2] => 1
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,5,1,2,3,4] => [3,4,5,6,2,1] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [5,4,6,1,2,3] => [4,5,6,2,1,3] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [5,4,1,6,2,3] => [3,5,6,2,1,4] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [4,3,5,6,1,2] => [5,6,2,1,3,4] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [4,3,6,5,1,2] => [5,6,2,1,4,3] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [5,4,1,2,6,3] => [3,4,6,2,1,5] => 1
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [4,3,6,1,5,2] => [4,6,2,1,5,3] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [4,3,1,5,6,2] => [3,6,2,1,4,5] => 1
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [4,3,1,6,5,2] => [3,6,2,1,5,4] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [3,2,4,5,6,1] => [6,2,1,3,4,5] => 0
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [3,2,5,4,6,1] => [6,2,1,4,3,5] => 0
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [3,2,5,6,4,1] => [6,2,1,5,3,4] => 0
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [3,2,6,4,5,1] => [6,2,1,4,5,3] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [3,2,6,5,4,1] => [6,2,1,5,4,3] => 0
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [5,4,1,2,3,6] => [3,4,5,2,1,6] => 1
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [4,3,6,1,2,5] => [4,5,2,1,6,3] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [4,3,1,6,2,5] => [3,5,2,1,6,4] => 2
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [3,2,5,6,1,4] => [5,2,1,6,3,4] => 1
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [3,2,6,5,1,4] => [5,2,1,6,4,3] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [4,3,1,2,5,6] => [3,4,2,1,5,6] => 1
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [3,2,6,1,4,5] => [4,2,1,5,6,3] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [4,3,1,2,6,5] => [3,4,2,1,6,5] => 1
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [3,2,6,1,5,4] => [4,2,1,6,5,3] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [3,2,1,5,4,6] => [3,2,1,5,4,6] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [3,2,1,5,6,4] => [3,2,1,6,4,5] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [3,2,1,6,4,5] => [3,2,1,5,6,4] => 1
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [3,2,1,6,5,4] => [3,2,1,6,5,4] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [2,1,4,3,5,6] => [2,1,4,3,5,6] => 0
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [2,1,4,5,3,6] => [2,1,5,3,4,6] => 0
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [2,1,5,3,4,6] => [2,1,4,5,3,6] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [2,1,5,4,3,6] => [2,1,5,4,3,6] => 0
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [2,1,4,5,6,3] => [2,1,6,3,4,5] => 0
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [2,1,5,4,6,3] => [2,1,6,4,3,5] => 0
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [2,1,5,6,3,4] => [2,1,5,6,3,4] => 1
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [2,1,5,6,4,3] => [2,1,6,5,3,4] => 0
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => [2,1,6,3,4,5] => [2,1,4,5,6,3] => 1
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Description
The tier of a permutation.
This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$.
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as OEIS:A122890 and OEIS:A158830 in the form of triangles read by rows, see [sec. 4, 1].
This is the number of elements $i$ such that $[i+1,k,i]$ is an occurrence of the pattern $[2,3,1]$. For example, $[3,5,6,1,2,4]$ has tier $2$, with witnesses $[3,5,2]$ (or $[3,6,2]$) and $[5,6,4]$.
According to [1], this is the number of passes minus one needed to sort the permutation using a single stack. The generating function for this statistic appears as OEIS:A122890 and OEIS:A158830 in the form of triangles read by rows, see [sec. 4, 1].
Map
inverse
Description
Sends a permutation to its inverse.
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 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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