- FindStat

Identifier

Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
St000769: Integer compositions ⟶ ℤ

Values

[1] => [1] => 0

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

[2] => [2] => 0

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

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

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

[3] => [3] => 0

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

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

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

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

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

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

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

[4] => [4] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[4,1] => [4,1] => 1

[5] => [5] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,4,1] => [4,1,1] => 1

[1,5] => [1,5] => 0

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

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

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

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

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

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

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

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

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

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

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

[3,3] => [3,3] => 0

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

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

[5,1] => [5,1] => 1

[6] => [6] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,4,1] => [4,1,1,1] => 1

[1,1,5] => [1,1,5] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,5,1] => [5,1,1] => 1

[1,6] => [1,6] => 0

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

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

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

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

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

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

>>> Load all 630 entries. <<<

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 3 + q$

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

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

$F_{6} = 11 + 8\ q + 7\ q^{2} + 4\ q^{3} + 2\ q^{4}$

$F_{7} = 15 + 15\ q + 14\ q^{2} + 10\ q^{3} + 6\ q^{4} + 3\ q^{5} + q^{6}$

$F_{8} = 22 + 23\ q + 26\ q^{2} + 21\ q^{3} + 17\ q^{4} + 10\ q^{5} + 6\ q^{6} + 2\ q^{7} + q^{8}$

$F_{9} = 30 + 37\ q + 44\ q^{2} + 42\ q^{3} + 36\ q^{4} + 27\ q^{5} + 19\ q^{6} + 11\ q^{7} + 6\ q^{8} + 3\ q^{9} + q^{10}$

Description

The major index of a composition regarded as a word.
This is the sum of the positions of the descents of the composition.
For the statistic which interprets the composition as a descent set, see St000008The major index of the composition..

Map

Foata bijection

Description

The Foata bijection for compositions.
The Foata bijection

$\phi$ is a bijection on the set of words whose letters are positive integers. 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 St000769The major index of a composition regarded as a word. to the number of inversions St000766The number of inversions of an integer composition..