- FindStat

Identifier

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001004: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

[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] => 4

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

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

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

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

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

[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] => 4

[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] => 3

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 5

[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] => 5

[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] => 4

[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] => 3

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

[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] => 4

[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] => 3

[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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of indices that are either left-to-right maxima or right-to-left minima.
The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a $321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see St000371The number of mid points of decreasing subsequences of length 3 in a permutation. and St000372The number of mid points of increasing subsequences of length 3 in a permutation..

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..