- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000018: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2 + q$

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

$F_{4} = 24 + 36\ q + 30\ q^{2} + 20\ q^{3} + 10\ q^{4} + 4\ q^{5} + q^{6}$

$F_{5} = 120 + 240\ q + 270\ q^{2} + 240\ q^{3} + 180\ q^{4} + 120\ q^{5} + 70\ q^{6} + 35\ q^{7} + 15\ q^{8} + 5\ q^{9} + q^{10}$

$F_{6} = 720 + 1800\ q + 2520\ q^{2} + 2730\ q^{3} + 2520\ q^{4} + 2100\ q^{5} + 1610\ q^{6} + 1140\ q^{7} + 750\ q^{8} + 455\ q^{9} + 252\ q^{10} + 126\ q^{11} + 56\ q^{12} + 21\ q^{13} + 6\ q^{14} + q^{15}$

Description

The number of inversions of a permutation.
This equals the minimal number of simple transpositions

$(i,i+1)$ needed to write

$\pi$ . Thus, it is also the Coxeter length 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..

Map

to Dyck path

Description

Sends a parking function to its support Dyck word.