Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000769
Mapping
St000769: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[2] => 0
[1,1,1] => 0
[1,2] => 0
[2,1] => 1
[3] => 0
[1,1,1,1] => 0
[1,1,2] => 0
[1,2,1] => 2
[1,3] => 0
[2,1,1] => 1
[2,2] => 0
[3,1] => 1
[4] => 0
[1,1,1,1,1] => 0
[1,1,1,2] => 0
[1,1,2,1] => 3
[1,1,3] => 0
[1,2,1,1] => 2
[1,2,2] => 0
[1,3,1] => 2
[1,4] => 0
[2,1,1,1] => 1
[2,1,2] => 1
[2,2,1] => 2
[2,3] => 0
[3,1,1] => 1
[3,2] => 1
[4,1] => 1
[5] => 0
[1,1,1,1,1,1] => 0
[1,1,1,1,2] => 0
[1,1,1,2,1] => 4
[1,1,1,3] => 0
[1,1,2,1,1] => 3
[1,1,2,2] => 0
[1,1,3,1] => 3
[1,1,4] => 0
[1,2,1,1,1] => 2
[1,2,1,2] => 2
[1,2,2,1] => 3
[1,2,3] => 0
[1,3,1,1] => 2
[1,3,2] => 2
[1,4,1] => 2
[1,5] => 0
[2,1,1,1,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 4
Description
The major index of a composition regarded as a word. This is the sum of the positions of the descents of the composition. For the statistic which interprets the composition as a descent set, see [[St000008]].
Matching statistic: St000766
(load all 2 compositions to match this statistic)
Mapping
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
St000766: Integer compositions ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 79%
Values
[1] => [1] => 0
[1,1] => [1,1] => 0
[2] => [2] => 0
[1,1,1] => [1,1,1] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[3] => [3] => 0
[1,1,1,1] => [1,1,1,1] => 0
[1,1,2] => [1,1,2] => 0
[1,2,1] => [2,1,1] => 2
[1,3] => [1,3] => 0
[2,1,1] => [1,2,1] => 1
[2,2] => [2,2] => 0
[3,1] => [3,1] => 1
[4] => [4] => 0
[1,1,1,1,1] => [1,1,1,1,1] => 0
[1,1,1,2] => [1,1,1,2] => 0
[1,1,2,1] => [2,1,1,1] => 3
[1,1,3] => [1,1,3] => 0
[1,2,1,1] => [1,2,1,1] => 2
[1,2,2] => [1,2,2] => 0
[1,3,1] => [3,1,1] => 2
[1,4] => [1,4] => 0
[2,1,1,1] => [1,1,2,1] => 1
[2,1,2] => [2,1,2] => 1
[2,2,1] => [2,2,1] => 2
[2,3] => [2,3] => 0
[3,1,1] => [1,3,1] => 1
[3,2] => [3,2] => 1
[4,1] => [4,1] => 1
[5] => [5] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 0
[1,1,1,1,2] => [1,1,1,1,2] => 0
[1,1,1,2,1] => [2,1,1,1,1] => 4
[1,1,1,3] => [1,1,1,3] => 0
[1,1,2,1,1] => [1,2,1,1,1] => 3
[1,1,2,2] => [1,1,2,2] => 0
[1,1,3,1] => [3,1,1,1] => 3
[1,1,4] => [1,1,4] => 0
[1,2,1,1,1] => [1,1,2,1,1] => 2
[1,2,1,2] => [2,1,1,2] => 2
[1,2,2,1] => [2,1,2,1] => 3
[1,2,3] => [1,2,3] => 0
[1,3,1,1] => [1,3,1,1] => 2
[1,3,2] => [3,1,2] => 2
[1,4,1] => [4,1,1] => 2
[1,5] => [1,5] => 0
[2,1,1,1,1] => [1,1,1,2,1] => 1
[2,1,1,2] => [1,2,1,2] => 1
[2,1,2,1] => [2,2,1,1] => 4
[1,1,1,1,1,1,1,1,2] => [1,1,1,1,1,1,1,1,2] => ? = 0
[1,1,1,1,2,1,1,2] => [1,2,1,1,1,1,1,2] => ? = 5
[1,1,1,1,2,2,1,1] => [1,2,1,1,1,1,2,1] => ? = 6
[1,1,1,1,5,1] => [5,1,1,1,1,1] => ? = 5
[1,1,2,1,1,1,1,2] => [1,1,1,2,1,1,1,2] => ? = 3
[1,1,2,1,1,2,1,1] => [1,2,1,2,1,1,1,1] => ? = 9
[1,1,2,2,1,1,1,1] => [1,1,1,2,1,1,2,1] => ? = 4
[1,1,3,1,1,1,2] => [1,1,1,3,1,1,2] => ? = 3
[1,1,3,1,1,3] => [1,3,1,1,1,3] => ? = 3
[1,1,3,2,1,2] => [2,3,1,1,1,2] => ? = 7
[1,1,7,1] => [7,1,1,1] => ? = 3
[1,2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,2,1,1] => ? = 2
[1,2,1,1,4,1] => [2,1,4,1,1,1] => ? = 7
[1,2,1,2,1,1,2] => [1,2,2,1,1,1,2] => ? = 6
[1,2,2,1,1,1,2] => [1,1,2,1,2,1,2] => ? = 3
[1,2,2,1,2,1,1] => [1,2,2,1,2,1,1] => ? = 8
[1,2,3,1,1,2] => [1,2,1,1,3,2] => ? = 3
[1,3,1,1,1,1,2] => [1,1,1,1,3,1,2] => ? = 2
[1,3,1,1,3,1] => [3,1,3,1,1,1] => ? = 7
[1,3,1,2,1,2] => [1,3,2,1,1,2] => ? = 6
[1,3,2,1,1,2] => [1,2,3,1,1,2] => ? = 5
[1,3,3,1,2] => [1,3,1,3,2] => ? = 3
[1,3,5,1] => [3,1,5,1] => ? = 3
[1,4,1,1,1,2] => [1,1,1,4,1,2] => ? = 2
[1,4,1,1,2,1] => [4,1,1,2,1,1] => ? = 7
[1,4,1,2,2] => [1,1,4,2,2] => ? = 2
[1,4,2,1,2] => [2,4,1,1,2] => ? = 5
[1,4,4,1] => [4,1,4,1] => ? = 3
[1,5,1,1,1,1] => [1,1,1,5,1,1] => ? = 2
[1,5,1,1,2] => [1,1,5,1,2] => ? = 2
[1,5,1,3] => [1,5,1,3] => ? = 2
[1,5,2,2] => [1,5,2,2] => ? = 2
[1,5,3,1] => [5,3,1,1] => ? = 5
[1,5,4] => [5,1,4] => ? = 2
[1,6,2,1] => [6,2,1,1] => ? = 5
[1,6,3] => [6,1,3] => ? = 2
[1,7,2] => [7,1,2] => ? = 2
[2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2,1] => ? = 1
[2,1,1,1,1,1,1,2] => [1,1,1,1,1,2,1,2] => ? = 1
[2,1,1,1,1,2,1,1] => [1,2,1,1,1,2,1,1] => ? = 7
[2,1,1,2,1,1,1,1] => [1,1,1,2,1,2,1,1] => ? = 5
[2,1,1,2,2,2] => [1,2,1,2,2,2] => ? = 1
[2,1,2,1,1,3] => [1,2,2,1,1,3] => ? = 4
[2,1,2,2,1,2] => [2,2,1,2,1,2] => ? = 5
[2,1,2,3,1,1] => [1,2,2,1,3,1] => ? = 5
[2,1,6,1] => [2,6,1,1] => ? = 4
[2,2,1,1,1,1,1,1] => [1,1,1,1,1,2,2,1] => ? = 2
[2,2,1,1,2,1,1] => [1,2,1,2,2,1,1] => ? = 7
[2,2,1,1,2,2] => [1,2,2,1,2,2] => ? = 2
[2,2,2,1,1,1,1] => [1,1,1,2,2,2,1] => ? = 3
Description
The number of inversions of an integer composition. This is the number of pairs $(i,j)$ such that $i < j$ and $c_i > c_j$.