Identifier
Values
[1,1] => [1,0,1,0] => [1,2] => 1
[2] => [1,1,0,0] => [2,1] => 1
[1,1,1] => [1,0,1,0,1,0] => [1,2,3] => 1
[1,2] => [1,0,1,1,0,0] => [1,3,2] => 2
[2,1] => [1,1,0,0,1,0] => [2,1,3] => 2
[3] => [1,1,1,0,0,0] => [3,2,1] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 3
[1,2,1] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 5
[1,3] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 3
[2,1,1] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 3
[2,2] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 5
[3,1] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 3
[4] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 9
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 6
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 9
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 16
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 11
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 4
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 11
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 16
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 9
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 6
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 9
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 4
[5] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => 14
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => 10
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => 19
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => 35
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => 26
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 10
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => 14
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => 40
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => 61
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 35
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => 26
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 40
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => 19
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => 5
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => 19
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => 40
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3,6,5,4] => 26
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => 35
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => 61
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => 40
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [2,1,6,5,4,3] => 14
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => 10
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => 26
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => 35
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [3,2,1,6,5,4] => 19
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => 10
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => 14
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [5,4,3,2,1,6] => 5
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => 6
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,4,6,5,7] => 20
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => 15
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,3,5,4,6,7] => 34
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,3,5,4,7,6] => 64
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,3,6,5,4,7] => 50
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => 20
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,4,3,5,6,7] => 34
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,4,3,5,7,6] => 99
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5,7] => 155
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,4,3,7,6,5] => 90
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,2,5,4,3,6,7] => 71
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,2,5,4,3,7,6] => 111
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,2,6,5,4,3,7] => 55
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => 15
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,2,4,5,6,7] => 20
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [1,3,2,4,5,7,6] => 78
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,2,4,6,5,7] => 169
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,2,4,7,6,5] => 111
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,4,6,7] => 155
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4,7,6] => 272
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,6,5,4,7] => 181
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,7,6,5,4] => 64
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [1,4,3,2,5,6,7] => 50
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,4,3,2,5,7,6] => 132
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,4,3,2,6,5,7] => 181
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,4,3,2,7,6,5] => 99
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Description
The number of permutations with the same descent word as the given permutation.
The descent word of a permutation is the binary word given by Mp00109descent word. For a given permutation, this statistic is the number of permutations with the same descent word, so the number of elements in the fiber of the map Mp00109descent word containing a given permutation.
This statistic appears as up-down analysis in statistical applications in genetics, see [1] and the references therein.
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
bounce path
Description
The bounce path determined by an integer composition.