Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St001744: 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] => [3,1,2] => 1
[[.,.],[.,.]] => [1,3,2] => [2,1,3] => 0
[[.,[.,.]],.] => [2,1,3] => [1,3,2] => 0
[[[.,.],.],.] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]] => [3,4,2,1] => [4,1,3,2] => 1
[.,[[.,.],[.,.]]] => [2,4,3,1] => [3,1,4,2] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [4,3,1,2] => 1
[.,[[[.,.],.],.]] => [2,3,4,1] => [3,4,1,2] => 1
[[.,.],[.,[.,.]]] => [1,4,3,2] => [2,1,4,3] => 0
[[.,.],[[.,.],.]] => [1,3,4,2] => [2,4,1,3] => 1
[[.,[.,.]],[.,.]] => [2,1,4,3] => [3,2,1,4] => 0
[[[.,.],.],[.,.]] => [1,2,4,3] => [2,3,1,4] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [1,4,3,2] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,3,4,2] => 0
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,2,4,3] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => [1,3,2,4] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [5,1,4,3,2] => 1
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [4,1,5,3,2] => 0
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [5,4,1,3,2] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [4,5,1,3,2] => 1
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [3,1,5,4,2] => 0
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [3,5,1,4,2] => 1
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [4,3,1,5,2] => 0
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [3,4,1,5,2] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [5,4,3,1,2] => 1
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [4,5,3,1,2] => 1
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [3,5,4,1,2] => 1
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [4,3,5,1,2] => 1
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [3,4,5,1,2] => 1
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [2,1,5,4,3] => 0
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [2,5,1,4,3] => 1
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [2,4,1,5,3] => 0
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [2,5,4,1,3] => 1
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [2,4,5,1,3] => 1
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [3,2,1,5,4] => 0
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [3,2,5,1,4] => 1
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [2,3,1,5,4] => 0
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [2,3,5,1,4] => 1
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [4,3,2,1,5] => 0
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [3,4,2,1,5] => 0
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [2,4,3,1,5] => 0
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [3,2,4,1,5] => 0
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [2,3,4,1,5] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,5,4,3,2] => 0
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,4,5,3,2] => 0
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [1,3,5,4,2] => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,4,3,5,2] => 0
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,3,4,5,2] => 0
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,2,5,4,3] => 0
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,2,4,5,3] => 0
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [1,3,2,5,4] => 0
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,3,5,4] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,4,3,2,5] => 0
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,3,4,2,5] => 0
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,2,4,3,5] => 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] => [6,5,4,3,2,1] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [6,1,5,4,3,2] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [5,1,6,4,3,2] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [6,5,1,4,3,2] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [5,6,1,4,3,2] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [4,1,6,5,3,2] => 0
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [4,6,1,5,3,2] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [5,4,1,6,3,2] => 0
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [4,5,1,6,3,2] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [6,5,4,1,3,2] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [5,6,4,1,3,2] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [4,6,5,1,3,2] => 1
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [5,4,6,1,3,2] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [4,5,6,1,3,2] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [3,1,6,5,4,2] => 0
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [3,6,1,5,4,2] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [3,5,1,6,4,2] => 0
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [3,6,5,1,4,2] => 1
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [3,5,6,1,4,2] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [4,3,1,6,5,2] => 0
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [4,3,6,1,5,2] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [3,4,1,6,5,2] => 0
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [3,4,6,1,5,2] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [5,4,3,1,6,2] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [4,5,3,1,6,2] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [3,5,4,1,6,2] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [4,3,5,1,6,2] => 0
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [3,4,5,1,6,2] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [6,5,4,3,1,2] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [5,6,4,3,1,2] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [4,6,5,3,1,2] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [5,4,6,3,1,2] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [4,5,6,3,1,2] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [3,6,5,4,1,2] => 1
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [3,5,6,4,1,2] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [4,3,6,5,1,2] => 1
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [3,4,6,5,1,2] => 1
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Description
The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map Mp00087inverse first fundamental transformation. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map Mp00087inverse first fundamental transformation. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
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.
Map
Tanimoto
Description
Add 1 to every entry of the permutation (n becomes 1 instead of n+1), except that when n appears at the front or the back of the permutation, instead remove it and place 1 at the other end of the permutation.
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