- FindStat

Identifier

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000446: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,3,5,6],[2,4]] => [2,4,1,3,5,6] => [2,1,4,3,5,6] => 8

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

[[1,3,4,6],[2,5]] => [2,5,1,3,4,6] => [2,1,3,5,4,6] => 7

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

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

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

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

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

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

[[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 9

[[1,3,5,6],[2],[4]] => [4,2,1,3,5,6] => [2,4,1,3,5,6] => 8

[[1,2,5,6],[3],[4]] => [4,3,1,2,5,6] => [1,4,3,2,5,6] => 7

[[1,3,4,6],[2],[5]] => [5,2,1,3,4,6] => [2,1,5,3,4,6] => 7

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

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

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

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

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

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

[[1,3,5],[2,4,6]] => [2,4,6,1,3,5] => [2,1,4,3,6,5] => 9

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

[[1,3,4],[2,5,6]] => [2,5,6,1,3,4] => [2,1,3,5,6,4] => 7

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

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

[[1,4,6],[2,5],[3]] => [3,2,5,1,4,6] => [3,2,1,5,4,6] => 11

[[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [2,4,1,5,3,6] => 8

[[1,2,6],[3,5],[4]] => [4,3,5,1,2,6] => [1,4,3,5,2,6] => 7

[[1,3,6],[2,4],[5]] => [5,2,4,1,3,6] => [2,1,5,4,3,6] => 10

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

[[1,4,5],[2,6],[3]] => [3,2,6,1,4,5] => [3,2,1,4,6,5] => 10

[[1,3,5],[2,6],[4]] => [4,2,6,1,3,5] => [2,4,1,3,6,5] => 9

[[1,2,5],[3,6],[4]] => [4,3,6,1,2,5] => [1,4,3,2,6,5] => 8

[[1,3,4],[2,6],[5]] => [5,2,6,1,3,4] => [2,1,5,3,6,4] => 7

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

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

[[1,3,5],[2,4],[6]] => [6,2,4,1,3,5] => [2,1,4,6,3,5] => 9

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

[[1,3,4],[2,5],[6]] => [6,2,5,1,3,4] => [2,1,3,6,5,4] => 8

[[1,2,4],[3,5],[6]] => [6,3,5,1,2,4] => [1,3,2,6,5,4] => 7

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

[[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 12

[[1,4,6],[2],[3],[5]] => [5,3,2,1,4,6] => [3,5,2,1,4,6] => 11

[[1,3,6],[2],[4],[5]] => [5,4,2,1,3,6] => [2,5,4,1,3,6] => 10

[[1,2,6],[3],[4],[5]] => [5,4,3,1,2,6] => [1,5,4,3,2,6] => 9

[[1,4,5],[2],[3],[6]] => [6,3,2,1,4,5] => [3,2,6,1,4,5] => 10

[[1,3,5],[2],[4],[6]] => [6,4,2,1,3,5] => [2,4,6,1,3,5] => 9

[[1,2,5],[3],[4],[6]] => [6,4,3,1,2,5] => [1,4,6,3,2,5] => 8

[[1,3,4],[2],[5],[6]] => [6,5,2,1,3,4] => [2,1,6,5,3,4] => 8

[[1,2,4],[3],[5],[6]] => [6,5,3,1,2,4] => [1,3,6,5,2,4] => 7

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

[[1,4],[2,5],[3,6]] => [3,6,2,5,1,4] => [3,2,1,6,5,4] => 12

[[1,3],[2,5],[4,6]] => [4,6,2,5,1,3] => [2,4,1,6,5,3] => 9

>>> Load all 224 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + q + q^{2} + q^{3}$

$F_{4} = 1 + q + 2\ q^{2} + 2\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

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

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

Description

The disorder of a permutation.
Consider a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ and cyclically scanning

$\pi$ from left to right and remove the elements

$1$ through

$n$ on this order one after the other. The disorder of

$\pi$ is defined to be the number of times a position was not removed in this process.
For example, the disorder of

$[3,5,2,1,4]$ is

$8$ since on the first scan, 3,5,2 and 4 are not removed, on the second, 3,5 and 4, and on the third and last scan, 5 is once again not removed.

Map

reading word permutation

Description

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.

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.).