- FindStat

Identifier

Mp00066: Permutations —inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001379: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

[3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 6

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

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

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

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

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

[1,4,3,2] => [1,4,3,2] => [4,3,1,2] => [3,4,2,1] => 10

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

[2,1,4,3] => [2,1,4,3] => [4,2,1,3] => [3,2,4,1] => 8

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

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

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

[2,4,3,1] => [4,1,3,2] => [4,1,3,2] => [2,4,3,1] => 9

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

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

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

[3,2,4,1] => [4,2,1,3] => [2,4,1,3] => [3,1,4,2] => 7

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

[3,4,2,1] => [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 8

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

[4,1,3,2] => [2,4,3,1] => [4,2,3,1] => [4,2,3,1] => 9

[4,2,1,3] => [3,2,4,1] => [3,2,4,1] => [4,2,1,3] => 7

[4,2,3,1] => [4,2,3,1] => [2,4,3,1] => [4,1,3,2] => 8

[4,3,1,2] => [3,4,2,1] => [3,4,2,1] => [4,3,1,2] => 8

[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 12

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

[1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 8

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

[1,2,4,5,3] => [1,2,5,3,4] => [1,5,2,3,4] => [1,3,4,5,2] => 7

[1,2,5,3,4] => [1,2,4,5,3] => [4,1,2,5,3] => [2,3,5,1,4] => 7

[1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => [3,4,5,2,1] => 14

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

[1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => [3,4,2,5,1] => 12

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

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

[1,3,5,2,4] => [1,4,2,5,3] => [4,5,1,2,3] => [3,4,5,1,2] => 9

[1,3,5,4,2] => [1,5,2,4,3] => [5,1,4,2,3] => [2,4,5,3,1] => 13

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

[1,4,2,5,3] => [1,3,5,2,4] => [3,1,2,5,4] => [2,3,1,5,4] => 9

[1,4,3,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => [3,4,2,1,5] => 10

[1,4,3,5,2] => [1,5,3,2,4] => [3,5,1,2,4] => [3,4,1,5,2] => 11

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

[1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => [1,4,5,3,2] => 12

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

[1,5,2,4,3] => [1,3,5,4,2] => [5,3,1,4,2] => [3,5,2,4,1] => 13

[1,5,3,2,4] => [1,4,3,5,2] => [4,3,1,5,2] => [3,5,2,1,4] => 11

[1,5,3,4,2] => [1,5,3,4,2] => [5,1,3,4,2] => [2,5,3,4,1] => 12

[1,5,4,2,3] => [1,4,5,3,2] => [4,5,1,3,2] => [3,5,4,1,2] => 12

[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => [4,5,3,2,1] => 18

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

[2,1,3,5,4] => [2,1,3,5,4] => [5,2,1,3,4] => [3,2,4,5,1] => 10

[2,1,4,3,5] => [2,1,4,3,5] => [4,2,1,3,5] => [3,2,4,1,5] => 8

[2,1,4,5,3] => [2,1,5,3,4] => [2,5,1,3,4] => [3,1,4,5,2] => 9

[2,1,5,3,4] => [2,1,4,5,3] => [4,2,1,5,3] => [3,2,5,1,4] => 9

[2,1,5,4,3] => [2,1,5,4,3] => [5,4,2,1,3] => [4,3,5,2,1] => 16

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

[2,3,1,5,4] => [3,1,2,5,4] => [5,1,3,2,4] => [2,4,3,5,1] => 11

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

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

[2,3,5,1,4] => [4,1,2,5,3] => [4,1,5,2,3] => [2,4,5,1,3] => 8

[2,3,5,4,1] => [5,1,2,4,3] => [5,1,2,4,3] => [2,3,5,4,1] => 12

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

[2,4,1,5,3] => [3,1,5,2,4] => [3,1,5,2,4] => [2,4,1,5,3] => 10

[2,4,3,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => [2,4,3,1,5] => 9

[2,4,3,5,1] => [5,1,3,2,4] => [1,5,3,2,4] => [1,4,3,5,2] => 10

[2,4,5,1,3] => [4,1,5,2,3] => [1,4,5,2,3] => [1,4,5,2,3] => 7

[2,4,5,3,1] => [5,1,4,2,3] => [1,5,2,4,3] => [1,3,5,4,2] => 11

[2,5,1,3,4] => [3,1,4,5,2] => [3,4,1,5,2] => [3,5,1,2,4] => 7

[2,5,1,4,3] => [3,1,5,4,2] => [5,3,4,1,2] => [4,5,2,3,1] => 14

[2,5,3,1,4] => [4,1,3,5,2] => [4,1,3,5,2] => [2,5,3,1,4] => 10

[2,5,3,4,1] => [5,1,3,4,2] => [3,1,5,4,2] => [2,5,1,4,3] => 11

[2,5,4,1,3] => [4,1,5,3,2] => [4,5,3,1,2] => [4,5,3,1,2] => 13

[2,5,4,3,1] => [5,1,4,3,2] => [5,4,1,3,2] => [3,5,4,2,1] => 17

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

[3,1,2,5,4] => [2,3,1,5,4] => [5,2,3,1,4] => [4,2,3,5,1] => 11

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

[3,1,4,5,2] => [2,5,1,3,4] => [2,1,3,5,4] => [2,1,3,5,4] => 7

[3,1,5,2,4] => [2,4,1,5,3] => [4,2,5,1,3] => [4,2,5,1,3] => 10

[3,1,5,4,2] => [2,5,1,4,3] => [5,2,1,4,3] => [3,2,5,4,1] => 14

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

[3,2,1,5,4] => [3,2,1,5,4] => [5,3,2,1,4] => [4,3,2,5,1] => 14

[3,2,4,1,5] => [4,2,1,3,5] => [2,4,1,3,5] => [3,1,4,2,5] => 7

[3,2,4,5,1] => [5,2,1,3,4] => [2,1,5,3,4] => [2,1,4,5,3] => 8

[3,2,5,1,4] => [4,2,1,5,3] => [4,5,2,1,3] => [4,3,5,1,2] => 11

[3,2,5,4,1] => [5,2,1,4,3] => [5,2,4,1,3] => [4,2,5,3,1] => 15

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

[3,4,1,5,2] => [3,5,1,2,4] => [1,3,2,5,4] => [1,3,2,5,4] => 8

[3,4,2,1,5] => [4,3,1,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 8

[3,4,2,5,1] => [5,3,1,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 9

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

[3,4,5,2,1] => [5,4,1,2,3] => [1,2,5,4,3] => [1,2,5,4,3] => 10

[3,5,1,2,4] => [3,4,1,5,2] => [3,4,5,1,2] => [4,5,1,2,3] => 8

[3,5,1,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => 12

>>> Load all 1200 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q^{2}$

$F_{3} = 1 + q^{2} + 2\ q^{3} + q^{4} + q^{6}$

$F_{4} = 1 + q^{2} + 2\ q^{3} + 4\ q^{4} + 2\ q^{5} + 4\ q^{6} + 2\ q^{7} + 4\ q^{8} + 2\ q^{9} + q^{10} + q^{12}$

$F_{5} = 1 + q^{2} + 2\ q^{3} + 4\ q^{4} + 6\ q^{5} + 7\ q^{6} + 8\ q^{7} + 12\ q^{8} + 12\ q^{9} + 14\ q^{10} + 12\ q^{11} + 12\ q^{12} + 8\ q^{13} + 7\ q^{14} + 6\ q^{15} + 4\ q^{16} + 2\ q^{17} + q^{18} + q^{20}$

$F_{6} = 1 + q^{2} + 2\ q^{3} + 4\ q^{4} + 6\ q^{5} + 12\ q^{6} + 12\ q^{7} + 21\ q^{8} + 26\ q^{9} + 37\ q^{10} + 40\ q^{11} + 55\ q^{12} + 52\ q^{13} + 61\ q^{14} + 60\ q^{15} + 61\ q^{16} + 52\ q^{17} + 55\ q^{18} + 40\ q^{19} + 37\ q^{20} + 26\ q^{21} + 21\ q^{22} + 12\ q^{23} + 12\ q^{24} + 6\ q^{25} + 4\ q^{26} + 2\ q^{27} + q^{28} + q^{30}$

Description

The number of inversions plus the major index of a permutation.
This is, the sum of St000004The major index of a permutation. and St000018The number of inversions of a permutation..

Map

inverse

Description

Sends a permutation to its inverse.

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.).