- FindStat

Identifier

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001759: 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] => 1

[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] => 2

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

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

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

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

[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] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

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

$F_{5} = 1 + q + 3\ q^{2} + 6\ q^{3} + 13\ q^{4} + 13\ q^{5} + 27\ q^{6} + 25\ q^{7} + 20\ q^{8} + 10\ q^{9} + q^{10}$

$F_{6} = 1 + q + 3\ q^{2} + 6\ q^{3} + 13\ q^{4} + 22\ q^{5} + 36\ q^{6} + 52\ q^{7} + 85\ q^{8} + 107\ q^{9} + 109\ q^{10} + 134\ q^{11} + 85\ q^{12} + 50\ q^{13} + 15\ q^{14} + q^{15}$

Description

The Rajchgot index of a permutation.
The Rajchgot index of a permutation

$\sigma$ is the degree of the Grothendieck polynomial of

$\sigma$ . This statistic on permutations was defined by Pechenik, Speyer, and Weigandt [1]. It can be computed by taking the maximum major index St000004The major index of a permutation. of the permutations smaller than or equal to

$\sigma$ in the right weak Bruhat order.

Map

inverse

Description

Sends a permutation to its inverse.

Map

Inverse fireworks map

Description

Sends a permutation to an inverse fireworks permutation.
A permutation

$\sigma$ is inverse fireworks if its inverse avoids the vincular pattern

$3-12$ . The inverse fireworks map sends any permutation

$\sigma$ to an inverse fireworks permutation that is below

$\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index 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.).