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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000472
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(load all 4 compositions to match this statistic)
St000472: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 1
[2,1] => 0
[1,2,3] => 3
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 6
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 4
[1,4,2,3] => 3
[1,4,3,2] => 1
[2,1,3,4] => 4
[2,1,4,3] => 1
[2,3,1,4] => 3
[2,3,4,1] => 5
[2,4,1,3] => 3
[2,4,3,1] => 2
[3,1,2,4] => 3
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 2
[3,4,1,2] => 4
[3,4,2,1] => 3
[4,1,2,3] => 3
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 2
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 10
[1,2,3,5,4] => 6
[1,2,4,3,5] => 6
[1,2,4,5,3] => 7
[1,2,5,3,4] => 6
[1,2,5,4,3] => 3
[1,3,2,4,5] => 7
[1,3,2,5,4] => 3
[1,3,4,2,5] => 6
[1,3,4,5,2] => 8
[1,3,5,2,4] => 6
[1,3,5,4,2] => 4
[1,4,2,3,5] => 6
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 4
[1,4,5,2,3] => 7
[1,4,5,3,2] => 5
Description
The sum of the ascent bottoms of a permutation.
Matching statistic: St000391
Mp00064: Permutations —reverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00131: Permutations —descent bottoms⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => 1 => 1
[2,1] => [1,2] => 0 => 0
[1,2,3] => [3,2,1] => 11 => 3
[1,3,2] => [2,3,1] => 10 => 1
[2,1,3] => [3,1,2] => 10 => 1
[2,3,1] => [1,3,2] => 01 => 2
[3,1,2] => [2,1,3] => 10 => 1
[3,2,1] => [1,2,3] => 00 => 0
[1,2,3,4] => [4,3,2,1] => 111 => 6
[1,2,4,3] => [3,4,2,1] => 110 => 3
[1,3,2,4] => [4,2,3,1] => 110 => 3
[1,3,4,2] => [2,4,3,1] => 101 => 4
[1,4,2,3] => [3,2,4,1] => 110 => 3
[1,4,3,2] => [2,3,4,1] => 100 => 1
[2,1,3,4] => [4,3,1,2] => 101 => 4
[2,1,4,3] => [3,4,1,2] => 100 => 1
[2,3,1,4] => [4,1,3,2] => 110 => 3
[2,3,4,1] => [1,4,3,2] => 011 => 5
[2,4,1,3] => [3,1,4,2] => 110 => 3
[2,4,3,1] => [1,3,4,2] => 010 => 2
[3,1,2,4] => [4,2,1,3] => 110 => 3
[3,1,4,2] => [2,4,1,3] => 100 => 1
[3,2,1,4] => [4,1,2,3] => 100 => 1
[3,2,4,1] => [1,4,2,3] => 010 => 2
[3,4,1,2] => [2,1,4,3] => 101 => 4
[3,4,2,1] => [1,2,4,3] => 001 => 3
[4,1,2,3] => [3,2,1,4] => 110 => 3
[4,1,3,2] => [2,3,1,4] => 100 => 1
[4,2,1,3] => [3,1,2,4] => 100 => 1
[4,2,3,1] => [1,3,2,4] => 010 => 2
[4,3,1,2] => [2,1,3,4] => 100 => 1
[4,3,2,1] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [5,4,3,2,1] => 1111 => 10
[1,2,3,5,4] => [4,5,3,2,1] => 1110 => 6
[1,2,4,3,5] => [5,3,4,2,1] => 1110 => 6
[1,2,4,5,3] => [3,5,4,2,1] => 1101 => 7
[1,2,5,3,4] => [4,3,5,2,1] => 1110 => 6
[1,2,5,4,3] => [3,4,5,2,1] => 1100 => 3
[1,3,2,4,5] => [5,4,2,3,1] => 1101 => 7
[1,3,2,5,4] => [4,5,2,3,1] => 1100 => 3
[1,3,4,2,5] => [5,2,4,3,1] => 1110 => 6
[1,3,4,5,2] => [2,5,4,3,1] => 1011 => 8
[1,3,5,2,4] => [4,2,5,3,1] => 1110 => 6
[1,3,5,4,2] => [2,4,5,3,1] => 1010 => 4
[1,4,2,3,5] => [5,3,2,4,1] => 1110 => 6
[1,4,2,5,3] => [3,5,2,4,1] => 1100 => 3
[1,4,3,2,5] => [5,2,3,4,1] => 1100 => 3
[1,4,3,5,2] => [2,5,3,4,1] => 1010 => 4
[1,4,5,2,3] => [3,2,5,4,1] => 1101 => 7
[1,4,5,3,2] => [2,3,5,4,1] => 1001 => 5
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000008
Mp00064: Permutations —reverse⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000008: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => 1 => [1,1] => 1
[2,1] => [1,2] => 0 => [2] => 0
[1,2,3] => [3,2,1] => 11 => [1,1,1] => 3
[1,3,2] => [2,3,1] => 10 => [1,2] => 1
[2,1,3] => [3,1,2] => 10 => [1,2] => 1
[2,3,1] => [1,3,2] => 01 => [2,1] => 2
[3,1,2] => [2,1,3] => 10 => [1,2] => 1
[3,2,1] => [1,2,3] => 00 => [3] => 0
[1,2,3,4] => [4,3,2,1] => 111 => [1,1,1,1] => 6
[1,2,4,3] => [3,4,2,1] => 110 => [1,1,2] => 3
[1,3,2,4] => [4,2,3,1] => 110 => [1,1,2] => 3
[1,3,4,2] => [2,4,3,1] => 101 => [1,2,1] => 4
[1,4,2,3] => [3,2,4,1] => 110 => [1,1,2] => 3
[1,4,3,2] => [2,3,4,1] => 100 => [1,3] => 1
[2,1,3,4] => [4,3,1,2] => 101 => [1,2,1] => 4
[2,1,4,3] => [3,4,1,2] => 100 => [1,3] => 1
[2,3,1,4] => [4,1,3,2] => 110 => [1,1,2] => 3
[2,3,4,1] => [1,4,3,2] => 011 => [2,1,1] => 5
[2,4,1,3] => [3,1,4,2] => 110 => [1,1,2] => 3
[2,4,3,1] => [1,3,4,2] => 010 => [2,2] => 2
[3,1,2,4] => [4,2,1,3] => 110 => [1,1,2] => 3
[3,1,4,2] => [2,4,1,3] => 100 => [1,3] => 1
[3,2,1,4] => [4,1,2,3] => 100 => [1,3] => 1
[3,2,4,1] => [1,4,2,3] => 010 => [2,2] => 2
[3,4,1,2] => [2,1,4,3] => 101 => [1,2,1] => 4
[3,4,2,1] => [1,2,4,3] => 001 => [3,1] => 3
[4,1,2,3] => [3,2,1,4] => 110 => [1,1,2] => 3
[4,1,3,2] => [2,3,1,4] => 100 => [1,3] => 1
[4,2,1,3] => [3,1,2,4] => 100 => [1,3] => 1
[4,2,3,1] => [1,3,2,4] => 010 => [2,2] => 2
[4,3,1,2] => [2,1,3,4] => 100 => [1,3] => 1
[4,3,2,1] => [1,2,3,4] => 000 => [4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => 1111 => [1,1,1,1,1] => 10
[1,2,3,5,4] => [4,5,3,2,1] => 1110 => [1,1,1,2] => 6
[1,2,4,3,5] => [5,3,4,2,1] => 1110 => [1,1,1,2] => 6
[1,2,4,5,3] => [3,5,4,2,1] => 1101 => [1,1,2,1] => 7
[1,2,5,3,4] => [4,3,5,2,1] => 1110 => [1,1,1,2] => 6
[1,2,5,4,3] => [3,4,5,2,1] => 1100 => [1,1,3] => 3
[1,3,2,4,5] => [5,4,2,3,1] => 1101 => [1,1,2,1] => 7
[1,3,2,5,4] => [4,5,2,3,1] => 1100 => [1,1,3] => 3
[1,3,4,2,5] => [5,2,4,3,1] => 1110 => [1,1,1,2] => 6
[1,3,4,5,2] => [2,5,4,3,1] => 1011 => [1,2,1,1] => 8
[1,3,5,2,4] => [4,2,5,3,1] => 1110 => [1,1,1,2] => 6
[1,3,5,4,2] => [2,4,5,3,1] => 1010 => [1,2,2] => 4
[1,4,2,3,5] => [5,3,2,4,1] => 1110 => [1,1,1,2] => 6
[1,4,2,5,3] => [3,5,2,4,1] => 1100 => [1,1,3] => 3
[1,4,3,2,5] => [5,2,3,4,1] => 1100 => [1,1,3] => 3
[1,4,3,5,2] => [2,5,3,4,1] => 1010 => [1,2,2] => 4
[1,4,5,2,3] => [3,2,5,4,1] => 1101 => [1,1,2,1] => 7
[1,4,5,3,2] => [2,3,5,4,1] => 1001 => [1,3,1] => 5
Description
The major index of the composition.
The descents of a composition $[c_1,c_2,\dots,c_k]$ are the partial sums $c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}$, excluding the sum of all parts. The major index of a composition is the sum of its descents.
For details about the major index see [[Permutations/Descents-Major]].
Matching statistic: St000154
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(load all 5 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
St000154: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 76%
St000154: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 76%
Values
[1,2] => [2,1] => 1
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 3
[1,3,2] => [2,3,1] => 1
[2,1,3] => [3,1,2] => 1
[2,3,1] => [1,3,2] => 2
[3,1,2] => [2,1,3] => 1
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 6
[1,2,4,3] => [3,4,2,1] => 3
[1,3,2,4] => [4,2,3,1] => 3
[1,3,4,2] => [2,4,3,1] => 4
[1,4,2,3] => [3,2,4,1] => 3
[1,4,3,2] => [2,3,4,1] => 1
[2,1,3,4] => [4,3,1,2] => 4
[2,1,4,3] => [3,4,1,2] => 1
[2,3,1,4] => [4,1,3,2] => 3
[2,3,4,1] => [1,4,3,2] => 5
[2,4,1,3] => [3,1,4,2] => 3
[2,4,3,1] => [1,3,4,2] => 2
[3,1,2,4] => [4,2,1,3] => 3
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [4,1,2,3] => 1
[3,2,4,1] => [1,4,2,3] => 2
[3,4,1,2] => [2,1,4,3] => 4
[3,4,2,1] => [1,2,4,3] => 3
[4,1,2,3] => [3,2,1,4] => 3
[4,1,3,2] => [2,3,1,4] => 1
[4,2,1,3] => [3,1,2,4] => 1
[4,2,3,1] => [1,3,2,4] => 2
[4,3,1,2] => [2,1,3,4] => 1
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => 10
[1,2,3,5,4] => [4,5,3,2,1] => 6
[1,2,4,3,5] => [5,3,4,2,1] => 6
[1,2,4,5,3] => [3,5,4,2,1] => 7
[1,2,5,3,4] => [4,3,5,2,1] => 6
[1,2,5,4,3] => [3,4,5,2,1] => 3
[1,3,2,4,5] => [5,4,2,3,1] => 7
[1,3,2,5,4] => [4,5,2,3,1] => 3
[1,3,4,2,5] => [5,2,4,3,1] => 6
[1,3,4,5,2] => [2,5,4,3,1] => 8
[1,3,5,2,4] => [4,2,5,3,1] => 6
[1,3,5,4,2] => [2,4,5,3,1] => 4
[1,4,2,3,5] => [5,3,2,4,1] => 6
[1,4,2,5,3] => [3,5,2,4,1] => 3
[1,4,3,2,5] => [5,2,3,4,1] => 3
[1,4,3,5,2] => [2,5,3,4,1] => 4
[1,4,5,2,3] => [3,2,5,4,1] => 7
[1,4,5,3,2] => [2,3,5,4,1] => 5
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 21
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 15
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 15
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 16
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 15
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => ? = 10
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 16
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 10
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => ? = 15
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 17
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 15
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => ? = 11
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => ? = 15
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => ? = 10
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => ? = 10
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => ? = 11
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 16
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => ? = 12
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 15
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => ? = 10
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => ? = 10
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => ? = 11
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => ? = 10
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => ? = 6
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => ? = 17
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => ? = 11
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => ? = 11
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => ? = 12
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => ? = 11
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => ? = 6
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => ? = 16
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => ? = 10
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => ? = 15
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => ? = 18
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => ? = 15
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => ? = 12
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => ? = 15
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => ? = 10
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => ? = 10
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => ? = 12
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => ? = 16
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => ? = 13
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => ? = 15
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => ? = 10
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => ? = 10
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => ? = 12
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => ? = 10
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => ? = 7
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => ? = 16
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => ? = 10
Description
The sum of the descent bottoms of a permutation.
This statistic is given by
$$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_{i+1}.$$
For the descent tops, see [[St000111]].
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