- FindStat

Identifier

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000673: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 293 entries. <<<

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

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

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

$F_{5} = 1 + 10\ q^{2} + 10\ q^{3} + 15\ q^{4} + 6\ q^{5}$

$F_{6} = 1 + 15\ q^{2} + 20\ q^{3} + 45\ q^{4} + 36\ q^{5} + 15\ q^{6}$

Description

The number of non-fixed points of a permutation.
In other words, this statistic is

$n$ minus the number of fixed points (St000022The number of fixed points of a permutation.) of

$\pi$ .

Map

to non-crossing permutation

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..