- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000054: Permutations ⟶ ℤ

Values

[.,.] => [1] => [1] => 1

[.,[.,.]] => [2,1] => [1,2] => 1

[[.,.],.] => [1,2] => [2,1] => 2

[.,[.,[.,.]]] => [3,2,1] => [1,2,3] => 1

[.,[[.,.],.]] => [2,3,1] => [1,3,2] => 1

[[.,.],[.,.]] => [1,3,2] => [2,3,1] => 2

[[.,[.,.]],.] => [2,1,3] => [3,1,2] => 3

[[[.,.],.],.] => [1,2,3] => [3,2,1] => 3

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,2,3,4] => 1

[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,2,4,3] => 1

[.,[[.,.],[.,.]]] => [2,4,3,1] => [1,3,4,2] => 1

[.,[[.,[.,.]],.]] => [3,2,4,1] => [1,4,2,3] => 1

[.,[[[.,.],.],.]] => [2,3,4,1] => [1,4,3,2] => 1

[[.,.],[.,[.,.]]] => [1,4,3,2] => [2,3,4,1] => 2

[[.,.],[[.,.],.]] => [1,3,4,2] => [2,4,3,1] => 2

[[.,[.,.]],[.,.]] => [2,1,4,3] => [3,4,1,2] => 3

[[[.,.],.],[.,.]] => [1,2,4,3] => [3,4,2,1] => 3

[[.,[.,[.,.]]],.] => [3,2,1,4] => [4,1,2,3] => 4

[[.,[[.,.],.]],.] => [2,3,1,4] => [4,1,3,2] => 4

[[[.,.],[.,.]],.] => [1,3,2,4] => [4,2,3,1] => 4

[[[.,[.,.]],.],.] => [2,1,3,4] => [4,3,1,2] => 4

[[[[.,.],.],.],.] => [1,2,3,4] => [4,3,2,1] => 4

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,2,3,4,5] => 1

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,2,3,5,4] => 1

[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [1,2,4,5,3] => 1

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [1,2,5,3,4] => 1

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,2,5,4,3] => 1

[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [1,3,4,5,2] => 1

[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [1,3,5,4,2] => 1

[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [1,4,5,2,3] => 1

[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [1,4,5,3,2] => 1

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [1,5,2,3,4] => 1

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,5,2,4,3] => 1

[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [1,5,3,4,2] => 1

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [1,5,4,2,3] => 1

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,5,4,3,2] => 1

[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [2,3,4,5,1] => 2

[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [2,3,5,4,1] => 2

[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [2,4,5,3,1] => 2

[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [2,5,3,4,1] => 2

[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [2,5,4,3,1] => 2

[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [3,4,5,1,2] => 3

[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [3,5,4,1,2] => 3

[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [3,4,5,2,1] => 3

[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [3,5,4,2,1] => 3

[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [4,5,1,2,3] => 4

[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [4,5,1,3,2] => 4

[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [4,5,2,3,1] => 4

[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [4,5,3,1,2] => 4

[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [4,5,3,2,1] => 4

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [5,1,2,3,4] => 5

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [5,1,2,4,3] => 5

[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [5,1,3,4,2] => 5

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [5,1,4,2,3] => 5

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [5,1,4,3,2] => 5

[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [5,2,3,4,1] => 5

[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [5,2,4,3,1] => 5

[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [5,3,4,1,2] => 5

[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [5,3,4,2,1] => 5

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [5,4,1,2,3] => 5

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [5,4,1,3,2] => 5

[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [5,4,2,3,1] => 5

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [5,4,3,1,2] => 5

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [5,4,3,2,1] => 5

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 1

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [1,2,3,4,6,5] => 1

[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [1,2,3,5,6,4] => 1

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [1,2,3,6,4,5] => 1

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [1,2,3,6,5,4] => 1

[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [1,2,4,5,6,3] => 1

[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [1,2,4,6,5,3] => 1

[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [1,2,5,6,3,4] => 1

[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [1,2,5,6,4,3] => 1

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [1,2,6,3,4,5] => 1

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [1,2,6,3,5,4] => 1

[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [1,2,6,4,5,3] => 1

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [1,2,6,5,3,4] => 1

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [1,2,6,5,4,3] => 1

[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [1,3,4,5,6,2] => 1

[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [1,3,4,6,5,2] => 1

[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [1,3,5,6,4,2] => 1

[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [1,3,6,4,5,2] => 1

[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [1,3,6,5,4,2] => 1

[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [1,4,5,6,2,3] => 1

[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [1,4,6,5,2,3] => 1

[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [1,4,5,6,3,2] => 1

[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [1,4,6,5,3,2] => 1

[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [1,5,6,2,3,4] => 1

[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [1,5,6,2,4,3] => 1

[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [1,5,6,3,4,2] => 1

[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [1,5,6,4,2,3] => 1

[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [1,5,6,4,3,2] => 1

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [1,6,2,3,4,5] => 1

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [1,6,2,3,5,4] => 1

[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [1,6,2,4,5,3] => 1

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [1,6,2,5,3,4] => 1

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [1,6,2,5,4,3] => 1

[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [1,6,3,4,5,2] => 1

[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [1,6,3,5,4,2] => 1

[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [1,6,4,5,2,3] => 1

[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [1,6,4,5,3,2] => 1

>>> Load all 392 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies St000703The number of deficiencies of a permutation. as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.

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.