Identifier
Values
[1] => [1,0] => [2,1] => [2,1] => 0
[1,1] => [1,0,1,0] => [3,1,2] => [1,3,2] => 1
[2] => [1,1,0,0] => [2,3,1] => [3,1,2] => 1
[1,1,1] => [1,0,1,0,1,0] => [4,1,2,3] => [1,2,4,3] => 2
[1,2] => [1,0,1,1,0,0] => [3,1,4,2] => [3,4,1,2] => 2
[2,1] => [1,1,0,0,1,0] => [2,4,1,3] => [1,3,4,2] => 2
[3] => [1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [1,2,3,5,4] => 3
[1,1,2] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [2,4,5,1,3] => 3
[1,2,1] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [3,1,4,5,2] => 3
[1,3] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,5,1,2,4] => 3
[2,1,1] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [1,3,2,5,4] => 2
[2,2] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [4,2,5,1,3] => 3
[3,1] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [1,3,4,5,2] => 3
[4] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 4
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [2,3,5,6,1,4] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [2,4,1,5,6,3] => 4
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [2,4,6,1,3,5] => 4
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [3,1,4,2,6,5] => 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [4,5,2,6,1,3] => 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [3,1,4,5,6,2] => 4
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,6,1,2,4,5] => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [1,3,2,4,6,5] => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [2,4,5,6,1,3] => 4
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [4,2,1,5,6,3] => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [4,2,6,1,3,5] => 4
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [1,3,4,2,6,5] => 3
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [5,2,3,6,1,4] => 4
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [1,3,4,5,6,2] => 4
[5] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => 4
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 5
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [2,3,4,6,7,1,5] => 5
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => 5
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 6
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [7,1,2,3,4,5,8,6] => [2,3,4,5,7,8,1,6] => 6
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => 6
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 7
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => [9,1,2,3,4,5,6,7,8] => 7
[1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [10,1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,10,9] => 8
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,1] => [10,1,2,3,4,5,6,7,8,9] => 8
[1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,11,10] => 9
[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,10,11,1] => [11,1,2,3,4,5,6,7,8,9,10] => 9
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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
bounce path
Description
The bounce path determined by an integer composition.
Map
major-index to inversion-number bijection
Description
Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.