Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000441
(load all 10 compositions to match this statistic)
Mapping
St000441: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 1
[2,1] => 0
[1,2,3] => 2
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 3
[1,2,4,3] => 1
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 2
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 0
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 2
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 4
[1,2,3,5,4] => 2
[1,2,4,3,5] => 1
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 1
[1,3,2,4,5] => 1
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 2
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 1
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 2
Description
The number of successions of a permutation. A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000214
(load all 10 compositions to match this statistic)
Mapping
Mp00069: Permutations —complement⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [2,1] => 1
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 2
[1,3,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => 0
[2,3,1] => [2,1,3] => 1
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 3
[1,2,4,3] => [4,3,1,2] => 1
[1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [4,2,1,3] => 1
[1,4,2,3] => [4,1,3,2] => 1
[1,4,3,2] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => 1
[2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [3,2,4,1] => 1
[2,3,4,1] => [3,2,1,4] => 2
[2,4,1,3] => [3,1,4,2] => 0
[2,4,3,1] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => 1
[3,1,4,2] => [2,4,1,3] => 0
[3,2,1,4] => [2,3,4,1] => 0
[3,2,4,1] => [2,3,1,4] => 0
[3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => 1
[4,1,2,3] => [1,4,3,2] => 2
[4,1,3,2] => [1,4,2,3] => 0
[4,2,1,3] => [1,3,4,2] => 0
[4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [1,2,4,3] => 1
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => 4
[1,2,3,5,4] => [5,4,3,1,2] => 2
[1,2,4,3,5] => [5,4,2,3,1] => 1
[1,2,4,5,3] => [5,4,2,1,3] => 2
[1,2,5,3,4] => [5,4,1,3,2] => 2
[1,2,5,4,3] => [5,4,1,2,3] => 1
[1,3,2,4,5] => [5,3,4,2,1] => 1
[1,3,2,5,4] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [5,3,2,4,1] => 1
[1,3,4,5,2] => [5,3,2,1,4] => 2
[1,3,5,2,4] => [5,3,1,4,2] => 0
[1,3,5,4,2] => [5,3,1,2,4] => 0
[1,4,2,3,5] => [5,2,4,3,1] => 1
[1,4,2,5,3] => [5,2,4,1,3] => 0
[1,4,3,2,5] => [5,2,3,4,1] => 0
[1,4,3,5,2] => [5,2,3,1,4] => 0
[1,4,5,2,3] => [5,2,1,4,3] => 2
[1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 3
[1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => ? = 4
[1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => ? = 4
[1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 3
[1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => ? = 3
[1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => ? = 4
[1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => ? = 2
[1,2,3,5,7,6,4] => [7,6,5,3,1,2,4] => ? = 2
[1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => ? = 3
[1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 2
[1,2,3,6,5,4,7] => [7,6,5,2,3,4,1] => ? = 2
[1,2,3,6,5,7,4] => [7,6,5,2,3,1,4] => ? = 2
[1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => ? = 4
[1,2,3,6,7,5,4] => [7,6,5,2,1,3,4] => ? = 3
[1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => ? = 4
[1,2,3,7,4,6,5] => [7,6,5,1,4,2,3] => ? = 2
[1,2,3,7,5,4,6] => [7,6,5,1,3,4,2] => ? = 2
[1,2,3,7,5,6,4] => [7,6,5,1,3,2,4] => ? = 3
[1,2,3,7,6,4,5] => [7,6,5,1,2,4,3] => ? = 3
[1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 3
[1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 1
[1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => ? = 1
[1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => ? = 2
[1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => ? = 2
[1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => ? = 3
[1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => ? = 2
[1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => ? = 3
[1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => ? = 4
[1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => ? = 2
[1,2,4,5,7,6,3] => [7,6,4,3,1,2,5] => ? = 2
[1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => ? = 1
[1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => ? = 1
[1,2,4,6,5,3,7] => [7,6,4,2,3,5,1] => ? = 1
[1,2,4,6,5,7,3] => [7,6,4,2,3,1,5] => ? = 1
[1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => ? = 2
[1,2,4,6,7,5,3] => [7,6,4,2,1,3,5] => ? = 2
[1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => ? = 2
[1,2,4,7,3,6,5] => [7,6,4,1,5,2,3] => ? = 1
[1,2,4,7,5,3,6] => [7,6,4,1,3,5,2] => ? = 1
[1,2,4,7,5,6,3] => [7,6,4,1,3,2,5] => ? = 2
[1,2,4,7,6,3,5] => [7,6,4,1,2,5,3] => ? = 1
[1,2,4,7,6,5,3] => [7,6,4,1,2,3,5] => ? = 1
[1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => ? = 3
[1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => ? = 2
[1,2,5,3,6,4,7] => [7,6,3,5,2,4,1] => ? = 1
[1,2,5,3,6,7,4] => [7,6,3,5,2,1,4] => ? = 2
[1,2,5,3,7,4,6] => [7,6,3,5,1,4,2] => ? = 1
[1,2,5,3,7,6,4] => [7,6,3,5,1,2,4] => ? = 1
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => ? = 2
[1,2,5,4,3,7,6] => [7,6,3,4,5,1,2] => ? = 1
Description
The number of adjacencies of a permutation. An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000237
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 91%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,1,2] => [2,3,1] => 2
[1,3,2] => [2,3,1] => [3,2,1] => [3,2,1] => 0
[2,1,3] => [3,1,2] => [2,3,1] => [3,1,2] => 0
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,1,2,3] => [2,3,4,1] => 3
[1,2,4,3] => [3,4,2,1] => [4,1,3,2] => [2,4,3,1] => 1
[1,3,2,4] => [4,2,3,1] => [4,3,1,2] => [3,4,2,1] => 0
[1,3,4,2] => [2,4,3,1] => [4,2,1,3] => [3,2,4,1] => 1
[1,4,2,3] => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,4,3,2] => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[2,1,3,4] => [4,3,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[2,1,4,3] => [3,4,1,2] => [2,4,3,1] => [4,1,3,2] => 0
[2,3,1,4] => [4,1,3,2] => [3,4,2,1] => [4,3,1,2] => 1
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [1,3,4,2] => 2
[2,4,1,3] => [3,1,4,2] => [3,4,1,2] => [3,4,1,2] => 0
[2,4,3,1] => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[3,1,2,4] => [4,2,1,3] => [3,1,4,2] => [2,4,1,3] => 1
[3,1,4,2] => [2,4,1,3] => [3,2,4,1] => [4,2,1,3] => 0
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,2,3] => [3,2,1,4] => [3,1,2,4] => [2,3,1,4] => 2
[4,1,3,2] => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 0
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,1,2,3,4] => [2,3,4,5,1] => 4
[1,2,3,5,4] => [4,5,3,2,1] => [5,1,2,4,3] => [2,3,5,4,1] => 2
[1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,2,3] => [2,4,5,3,1] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [5,1,3,2,4] => [2,4,3,5,1] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [5,1,4,3,2] => [2,5,4,3,1] => 2
[1,2,5,4,3] => [3,4,5,2,1] => [5,1,3,4,2] => [2,5,3,4,1] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,1,2,4] => [3,4,2,5,1] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [5,3,1,4,2] => [3,5,2,4,1] => 0
[1,3,4,2,5] => [5,2,4,3,1] => [5,4,1,3,2] => [3,5,4,2,1] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [5,2,1,3,4] => [3,2,4,5,1] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [5,4,1,2,3] => [3,4,5,2,1] => 0
[1,3,5,4,2] => [2,4,5,3,1] => [5,2,1,4,3] => [3,2,5,4,1] => 0
[1,4,2,3,5] => [5,3,2,4,1] => [5,4,2,1,3] => [4,3,5,2,1] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [5,4,3,1,2] => [4,5,3,2,1] => 0
[1,4,3,2,5] => [5,2,3,4,1] => [5,3,4,1,2] => [4,5,2,3,1] => 0
[1,4,3,5,2] => [2,5,3,4,1] => [5,2,4,1,3] => [4,2,5,3,1] => 0
[1,4,5,2,3] => [3,2,5,4,1] => [5,3,2,1,4] => [4,3,2,5,1] => 2
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [7,1,2,3,4,6,5] => [2,3,4,5,7,6,1] => ? = 4
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,1,2,3,6,4,5] => [2,3,4,6,7,5,1] => ? = 3
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [7,1,2,3,5,4,6] => [2,3,4,6,5,7,1] => ? = 4
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [7,1,2,3,6,5,4] => [2,3,4,7,6,5,1] => ? = 4
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [7,1,2,3,5,6,4] => [2,3,4,7,5,6,1] => ? = 3
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [7,1,2,5,3,4,6] => [2,3,5,6,4,7,1] => ? = 3
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [7,1,2,5,3,6,4] => [2,3,5,7,4,6,1] => ? = 2
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [7,1,2,6,3,5,4] => [2,3,5,7,6,4,1] => ? = 3
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [7,1,2,4,3,5,6] => [2,3,5,4,6,7,1] => ? = 4
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [7,1,2,6,3,4,5] => [2,3,5,6,7,4,1] => ? = 2
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [7,1,2,4,3,6,5] => [2,3,5,4,7,6,1] => ? = 2
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [7,1,2,6,4,3,5] => [2,3,6,5,7,4,1] => ? = 3
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [7,1,2,6,5,3,4] => [2,3,6,7,5,4,1] => ? = 2
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [7,1,2,5,6,3,4] => [2,3,6,7,4,5,1] => ? = 2
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [7,1,2,4,6,3,5] => [2,3,6,4,7,5,1] => ? = 2
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [7,1,2,5,4,3,6] => [2,3,6,5,4,7,1] => ? = 4
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [7,1,2,4,5,3,6] => [2,3,6,4,5,7,1] => ? = 3
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [7,1,2,6,4,5,3] => [2,3,7,5,6,4,1] => ? = 4
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [7,1,2,6,5,4,3] => [2,3,7,6,5,4,1] => ? = 2
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [7,1,2,5,6,4,3] => [2,3,7,6,4,5,1] => ? = 2
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [7,1,2,4,6,5,3] => [2,3,7,4,6,5,1] => ? = 3
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [7,1,2,5,4,6,3] => [2,3,7,5,4,6,1] => ? = 3
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => [2,3,7,4,5,6,1] => ? = 2
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [7,1,4,2,3,5,6] => [2,4,5,3,6,7,1] => ? = 3
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [7,1,4,2,3,6,5] => [2,4,5,3,7,6,1] => ? = 1
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [7,1,4,2,6,3,5] => [2,4,6,3,7,5,1] => ? = 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [7,1,4,2,5,3,6] => [2,4,6,3,5,7,1] => ? = 2
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [7,1,4,2,6,5,3] => [2,4,7,3,6,5,1] => ? = 2
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [7,1,4,2,5,6,3] => [2,4,7,3,5,6,1] => ? = 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [7,1,5,2,4,3,6] => [2,4,6,5,3,7,1] => ? = 3
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [7,1,5,2,4,6,3] => [2,4,7,5,3,6,1] => ? = 2
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [7,1,6,2,4,5,3] => [2,4,7,5,6,3,1] => ? = 3
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [7,1,3,2,4,5,6] => [2,4,3,5,6,7,1] => ? = 4
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [7,1,6,2,4,3,5] => [2,4,6,5,7,3,1] => ? = 2
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [7,1,3,2,4,6,5] => [2,4,3,5,7,6,1] => ? = 2
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [7,1,6,2,3,4,5] => [2,4,5,6,7,3,1] => ? = 1
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [7,1,6,2,5,4,3] => [2,4,7,6,5,3,1] => ? = 1
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [7,1,5,2,6,4,3] => [2,4,7,6,3,5,1] => ? = 1
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [7,1,3,2,6,4,5] => [2,4,3,6,7,5,1] => ? = 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [7,1,5,2,3,4,6] => [2,4,5,6,3,7,1] => ? = 2
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [7,1,3,2,5,4,6] => [2,4,3,6,5,7,1] => ? = 2
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [7,1,6,2,3,5,4] => [2,4,5,7,6,3,1] => ? = 2
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [7,1,6,2,5,3,4] => [2,4,6,7,5,3,1] => ? = 1
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [7,1,5,2,6,3,4] => [2,4,6,7,3,5,1] => ? = 1
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [7,1,3,2,6,5,4] => [2,4,3,7,6,5,1] => ? = 2
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [7,1,5,2,3,6,4] => [2,4,5,7,3,6,1] => ? = 1
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [7,1,3,2,5,6,4] => [2,4,3,7,5,6,1] => ? = 1
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [7,1,5,3,2,4,6] => [2,5,4,6,3,7,1] => ? = 3
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [7,1,5,3,2,6,4] => [2,5,4,7,3,6,1] => ? = 2
[1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [7,1,5,6,2,3,4] => [2,5,6,7,3,4,1] => ? = 1
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.