- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000672: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => [6,5,4,3,2,1,7] => 1

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,7,6,5,4,3,2] => 1

[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [7,6,5,4,3,2,1,8] => 1

[6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [6,5,3,4,2,1,7] => 2

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

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

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

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

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

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

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

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

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

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

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

[2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,7,6,4,5,3,2] => 2

[1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,8,7,6,5,4,3,2] => 1

[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [8,7,6,5,4,3,2,1,9] => 1

[7,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [7,6,4,5,3,2,1,8] => 2

[6,2] => [1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [6,4,3,5,2,1,7] => 2

[6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [6,3,5,4,2,1,7] => 2

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

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

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

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

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

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

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

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

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

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

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

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

[3,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,7,5,4,6,3,2] => 2

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

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

[2,2,1,1,1,1] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [1,7,4,6,5,3,2] => 2

[2,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1,8,7,5,6,4,3,2] => 2

[1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,9,8,7,6,5,4,3,2] => 1

[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [9,8,7,6,5,4,3,2,1,10] => 1

[8,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0] => [8,7,6,4,5,3,2,1,9] => 2

[7,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [7,6,4,3,5,2,1,8] => 2

[7,1,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [7,6,3,5,4,2,1,8] => 2

[6,3] => [1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [6,4,3,2,5,1,7] => 2

[6,2,1] => [1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [6,3,4,5,2,1,7] => 3

[6,1,1,1] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [6,2,5,4,3,1,7] => 2

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

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

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

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

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

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

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

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

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

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

[4,1,1,1,1,1] => [1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [1,7,5,4,3,6,2] => 2

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

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

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

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

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

[3,2,1,1,1,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [1,7,4,5,6,3,2] => 3

[3,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [1,8,7,5,4,6,3,2] => 2

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

[2,2,2,1,1,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [1,7,3,6,5,4,2] => 2

[2,2,1,1,1,1,1] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,8,7,4,6,5,3,2] => 2

[2,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1,9,8,7,5,6,4,3,2] => 2

[1,1,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,10,9,8,7,6,5,4,3,2] => 1

[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [10,9,8,7,6,5,4,3,2,1,11] => 1

[9,1] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0] => [9,8,7,5,6,4,3,2,1,10] => 2

[8,2] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0] => [8,7,5,4,6,3,2,1,9] => 2

[8,1,1] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0] => [8,7,4,6,5,3,2,1,9] => 2

[7,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [7,5,4,3,6,2,1,8] => 2

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Description

The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].

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 partition to the shortest Dyck path tracing the shape of its Ferrers diagram.