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Identifier

Mp00067: Permutations —Foata bijection⟶ Permutations

Images

[] => []

[1] => [1]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.).

References

[1] Foata, D., Schützenberger, M.-P. Major index and inversion number of permutations MathSciNet:0506852
[2] Foata bijection Mp00096

Properties

bijective, graded

Sage code

def mapping(elt):
    return elt.foata_bijection()

Weight

Created

Jan 26, 2020 at 12:32 by FindStatCrew

Updated

Jan 26, 2020 at 12:32 by Martin Rubey