- 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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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.