- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000308: Permutations ⟶ ℤ

Values

[1,0] => [1] => [1] => 1

[1,0,1,0] => [1,2] => [1,2] => 2

[1,1,0,0] => [2,1] => [2,1] => 1

[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 3

[1,0,1,1,0,0] => [1,3,2] => [3,1,2] => 2

[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => 2

[1,1,0,1,0,0] => [2,3,1] => [2,3,1] => 2

[1,1,1,0,0,0] => [3,2,1] => [3,2,1] => 1

[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 4

[1,0,1,0,1,1,0,0] => [1,2,4,3] => [4,1,2,3] => 3

[1,0,1,1,0,0,1,0] => [1,3,2,4] => [3,1,2,4] => 3

[1,0,1,1,0,1,0,0] => [1,3,4,2] => [3,1,4,2] => 2

[1,0,1,1,1,0,0,0] => [1,4,3,2] => [4,3,1,2] => 2

[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => 3

[1,1,0,0,1,1,0,0] => [2,1,4,3] => [4,2,1,3] => 2

[1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,3,1,4] => 2

[1,1,0,1,0,1,0,0] => [2,3,4,1] => [2,3,4,1] => 3

[1,1,0,1,1,0,0,0] => [2,4,3,1] => [4,2,3,1] => 2

[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1,4] => 2

[1,1,1,0,0,1,0,0] => [3,2,4,1] => [3,2,4,1] => 2

[1,1,1,0,1,0,0,0] => [3,4,2,1] => [3,4,2,1] => 2

[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,2,1] => 1

[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 5

[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [5,1,2,3,4] => 4

[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [4,1,2,3,5] => 4

[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [4,1,2,5,3] => 3

[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [5,4,1,2,3] => 3

[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [3,1,2,4,5] => 4

[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [5,3,1,2,4] => 3

[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [3,1,4,2,5] => 3

[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [3,1,4,5,2] => 3

[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [5,3,1,4,2] => 2

[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [4,3,1,2,5] => 3

[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [4,3,1,5,2] => 2

[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [4,5,1,3,2] => 2

[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [5,4,3,1,2] => 2

[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => 4

[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [5,2,1,3,4] => 3

[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [4,2,1,3,5] => 3

[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [4,2,1,5,3] => 2

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [5,4,2,1,3] => 2

[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [2,3,1,4,5] => 3

[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [5,2,3,1,4] => 2

[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [2,3,4,1,5] => 3

[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [2,3,4,5,1] => 4

[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [5,2,3,4,1] => 3

[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [4,2,3,1,5] => 2

[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [4,2,3,5,1] => 3

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [4,2,5,3,1] => 2

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [5,4,2,3,1] => 2

[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,2,1,4,5] => 3

[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [5,3,2,1,4] => 2

[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [3,2,4,1,5] => 2

[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [3,2,4,5,1] => 3

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [5,3,2,4,1] => 2

[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [3,4,2,1,5] => 2

[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [3,4,2,5,1] => 2

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [3,4,5,2,1] => 3

[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [5,3,4,2,1] => 2

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1,5] => 2

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [4,3,2,5,1] => 2

[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [4,3,5,2,1] => 2

[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [4,5,3,2,1] => 2

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,4,3,2,1] => 1

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 5

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 5

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [5,1,2,3,6,4] => 4

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => 4

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => 5

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => 4

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [4,1,2,5,3,6] => 4

[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => 4

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [6,4,1,2,5,3] => 3

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [5,4,1,2,3,6] => 4

[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [5,4,1,2,6,3] => 3

[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [5,6,1,2,4,3] => 3

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => 3

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 5

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [6,3,1,2,4,5] => 4

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [5,3,1,2,4,6] => 4

[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [5,3,1,2,6,4] => 3

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [6,5,3,1,2,4] => 3

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [3,1,4,2,5,6] => 4

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [6,3,1,4,2,5] => 3

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [3,1,4,5,2,6] => 3

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [3,1,4,5,6,2] => 4

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [6,3,1,4,5,2] => 3

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [5,3,1,4,2,6] => 3

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [5,3,1,4,6,2] => 3

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [5,3,1,6,4,2] => 2

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [6,5,3,1,4,2] => 2

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [4,3,1,2,5,6] => 4

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [6,4,3,1,2,5] => 3

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [4,3,1,5,2,6] => 3

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [4,3,1,5,6,2] => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [6,4,3,1,5,2] => 2

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [4,5,1,3,2,6] => 3

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [4,5,1,3,6,2] => 3

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [4,5,1,6,3,2] => 2

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [6,4,5,1,3,2] => 2

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = q + 3\ q^{2} + q^{3}$

$F_{4} = q + 8\ q^{2} + 4\ q^{3} + q^{4}$

$F_{5} = q + 20\ q^{2} + 15\ q^{3} + 5\ q^{4} + q^{5}$

$F_{6} = q + 50\ q^{2} + 52\ q^{3} + 22\ q^{4} + 6\ q^{5} + q^{6}$

Description

The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also St000325The width of the tree associated to a permutation. for the width of this tree.

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).

Map

Foata bijection

Description

Sends a permutation to its image under the Foata bijection.
The Foata bijection

$\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word

$w_1 w_2 ... w_n$ , compute the image inductively by starting with

$\phi(w_1) = w_1$ .
At the

$i$ -th step, if

$\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$ , define

$\phi(w_1 w_2 ... w_i w_{i+1})$ by placing

$w_{i+1}$ on the end of the word

$v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.

If $w_{i+1} \geq v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$ .
If $w_{i+1} < v_i$ , place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$ .

In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute

$\phi([1,4,2,5,3])$ , the sequence of words is

$1$
$|1|4 \to 14$
$|14|2 \to 412$
$|4|1|2|5 \to 4125$
$|4|125|3 \to 45123.$

In total, this gives

$\phi([1,4,2,5,3]) = [4,5,1,2,3]$ .
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).