- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000991: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 196 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

$F_{1} = q$

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

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

$F_{4} = q + 7\ q^{2} + 5\ q^{3} + q^{4}$

$F_{5} = q + 16\ q^{2} + 17\ q^{3} + 7\ q^{4} + q^{5}$

$F_{6} = q + 39\ q^{2} + 51\ q^{3} + 31\ q^{4} + 9\ q^{5} + q^{6}$

Description

The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see St000314The number of left-to-right-maxima of a permutation..

Map

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.

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.).