Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000111
(load all 13 compositions to match this statistic)
Mapping
St000111: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 2
[1,2,3] => 0
[1,3,2] => 3
[2,1,3] => 2
[2,3,1] => 3
[3,1,2] => 3
[3,2,1] => 5
[1,2,3,4] => 0
[1,2,4,3] => 4
[1,3,2,4] => 3
[1,3,4,2] => 4
[1,4,2,3] => 4
[1,4,3,2] => 7
[2,1,3,4] => 2
[2,1,4,3] => 6
[2,3,1,4] => 3
[2,3,4,1] => 4
[2,4,1,3] => 4
[2,4,3,1] => 7
[3,1,2,4] => 3
[3,1,4,2] => 7
[3,2,1,4] => 5
[3,2,4,1] => 7
[3,4,1,2] => 4
[3,4,2,1] => 6
[4,1,2,3] => 4
[4,1,3,2] => 7
[4,2,1,3] => 6
[4,2,3,1] => 7
[4,3,1,2] => 7
[4,3,2,1] => 9
[1,2,3,4,5] => 0
[1,2,3,5,4] => 5
[1,2,4,3,5] => 4
[1,2,4,5,3] => 5
[1,2,5,3,4] => 5
[1,2,5,4,3] => 9
[1,3,2,4,5] => 3
[1,3,2,5,4] => 8
[1,3,4,2,5] => 4
[1,3,4,5,2] => 5
[1,3,5,2,4] => 5
[1,3,5,4,2] => 9
[1,4,2,3,5] => 4
[1,4,2,5,3] => 9
[1,4,3,2,5] => 7
[1,4,3,5,2] => 9
[1,4,5,2,3] => 5
Description
The sum of the descent tops (or Genocchi descents) of a permutation. This statistic is given by $$\pi \mapsto \sum_{i\in\operatorname{Des}(\pi)} \pi_i.$$
Matching statistic: St000391
Mapping
Mp00130: Permutations —descent tops⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000391: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => 1 => 1 = 0 + 1
[1,2] => 0 => [2] => 10 => 1 = 0 + 1
[2,1] => 1 => [1,1] => 11 => 3 = 2 + 1
[1,2,3] => 00 => [3] => 100 => 1 = 0 + 1
[1,3,2] => 01 => [2,1] => 101 => 4 = 3 + 1
[2,1,3] => 10 => [1,2] => 110 => 3 = 2 + 1
[2,3,1] => 01 => [2,1] => 101 => 4 = 3 + 1
[3,1,2] => 01 => [2,1] => 101 => 4 = 3 + 1
[3,2,1] => 11 => [1,1,1] => 111 => 6 = 5 + 1
[1,2,3,4] => 000 => [4] => 1000 => 1 = 0 + 1
[1,2,4,3] => 001 => [3,1] => 1001 => 5 = 4 + 1
[1,3,2,4] => 010 => [2,2] => 1010 => 4 = 3 + 1
[1,3,4,2] => 001 => [3,1] => 1001 => 5 = 4 + 1
[1,4,2,3] => 001 => [3,1] => 1001 => 5 = 4 + 1
[1,4,3,2] => 011 => [2,1,1] => 1011 => 8 = 7 + 1
[2,1,3,4] => 100 => [1,3] => 1100 => 3 = 2 + 1
[2,1,4,3] => 101 => [1,2,1] => 1101 => 7 = 6 + 1
[2,3,1,4] => 010 => [2,2] => 1010 => 4 = 3 + 1
[2,3,4,1] => 001 => [3,1] => 1001 => 5 = 4 + 1
[2,4,1,3] => 001 => [3,1] => 1001 => 5 = 4 + 1
[2,4,3,1] => 011 => [2,1,1] => 1011 => 8 = 7 + 1
[3,1,2,4] => 010 => [2,2] => 1010 => 4 = 3 + 1
[3,1,4,2] => 011 => [2,1,1] => 1011 => 8 = 7 + 1
[3,2,1,4] => 110 => [1,1,2] => 1110 => 6 = 5 + 1
[3,2,4,1] => 011 => [2,1,1] => 1011 => 8 = 7 + 1
[3,4,1,2] => 001 => [3,1] => 1001 => 5 = 4 + 1
[3,4,2,1] => 101 => [1,2,1] => 1101 => 7 = 6 + 1
[4,1,2,3] => 001 => [3,1] => 1001 => 5 = 4 + 1
[4,1,3,2] => 011 => [2,1,1] => 1011 => 8 = 7 + 1
[4,2,1,3] => 101 => [1,2,1] => 1101 => 7 = 6 + 1
[4,2,3,1] => 011 => [2,1,1] => 1011 => 8 = 7 + 1
[4,3,1,2] => 011 => [2,1,1] => 1011 => 8 = 7 + 1
[4,3,2,1] => 111 => [1,1,1,1] => 1111 => 10 = 9 + 1
[1,2,3,4,5] => 0000 => [5] => 10000 => 1 = 0 + 1
[1,2,3,5,4] => 0001 => [4,1] => 10001 => 6 = 5 + 1
[1,2,4,3,5] => 0010 => [3,2] => 10010 => 5 = 4 + 1
[1,2,4,5,3] => 0001 => [4,1] => 10001 => 6 = 5 + 1
[1,2,5,3,4] => 0001 => [4,1] => 10001 => 6 = 5 + 1
[1,2,5,4,3] => 0011 => [3,1,1] => 10011 => 10 = 9 + 1
[1,3,2,4,5] => 0100 => [2,3] => 10100 => 4 = 3 + 1
[1,3,2,5,4] => 0101 => [2,2,1] => 10101 => 9 = 8 + 1
[1,3,4,2,5] => 0010 => [3,2] => 10010 => 5 = 4 + 1
[1,3,4,5,2] => 0001 => [4,1] => 10001 => 6 = 5 + 1
[1,3,5,2,4] => 0001 => [4,1] => 10001 => 6 = 5 + 1
[1,3,5,4,2] => 0011 => [3,1,1] => 10011 => 10 = 9 + 1
[1,4,2,3,5] => 0010 => [3,2] => 10010 => 5 = 4 + 1
[1,4,2,5,3] => 0011 => [3,1,1] => 10011 => 10 = 9 + 1
[1,4,3,2,5] => 0110 => [2,1,2] => 10110 => 8 = 7 + 1
[1,4,3,5,2] => 0011 => [3,1,1] => 10011 => 10 = 9 + 1
[1,4,5,2,3] => 0001 => [4,1] => 10001 => 6 = 5 + 1
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000471
(load all 5 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
St000471: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ? = 0
[1,2] => [2,1] => 0
[2,1] => [1,2] => 2
[1,2,3] => [3,2,1] => 0
[1,3,2] => [2,3,1] => 3
[2,1,3] => [3,1,2] => 2
[2,3,1] => [1,3,2] => 3
[3,1,2] => [2,1,3] => 3
[3,2,1] => [1,2,3] => 5
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => 4
[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] => 4
[1,4,3,2] => [2,3,4,1] => 7
[2,1,3,4] => [4,3,1,2] => 2
[2,1,4,3] => [3,4,1,2] => 6
[2,3,1,4] => [4,1,3,2] => 3
[2,3,4,1] => [1,4,3,2] => 4
[2,4,1,3] => [3,1,4,2] => 4
[2,4,3,1] => [1,3,4,2] => 7
[3,1,2,4] => [4,2,1,3] => 3
[3,1,4,2] => [2,4,1,3] => 7
[3,2,1,4] => [4,1,2,3] => 5
[3,2,4,1] => [1,4,2,3] => 7
[3,4,1,2] => [2,1,4,3] => 4
[3,4,2,1] => [1,2,4,3] => 6
[4,1,2,3] => [3,2,1,4] => 4
[4,1,3,2] => [2,3,1,4] => 7
[4,2,1,3] => [3,1,2,4] => 6
[4,2,3,1] => [1,3,2,4] => 7
[4,3,1,2] => [2,1,3,4] => 7
[4,3,2,1] => [1,2,3,4] => 9
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [4,5,3,2,1] => 5
[1,2,4,3,5] => [5,3,4,2,1] => 4
[1,2,4,5,3] => [3,5,4,2,1] => 5
[1,2,5,3,4] => [4,3,5,2,1] => 5
[1,2,5,4,3] => [3,4,5,2,1] => 9
[1,3,2,4,5] => [5,4,2,3,1] => 3
[1,3,2,5,4] => [4,5,2,3,1] => 8
[1,3,4,2,5] => [5,2,4,3,1] => 4
[1,3,4,5,2] => [2,5,4,3,1] => 5
[1,3,5,2,4] => [4,2,5,3,1] => 5
[1,3,5,4,2] => [2,4,5,3,1] => 9
[1,4,2,3,5] => [5,3,2,4,1] => 4
[1,4,2,5,3] => [3,5,2,4,1] => 9
[1,4,3,2,5] => [5,2,3,4,1] => 7
[1,4,3,5,2] => [2,5,3,4,1] => 9
[1,4,5,2,3] => [3,2,5,4,1] => 5
[1,4,5,3,2] => [2,3,5,4,1] => 8
Description
The sum of the ascent tops of a permutation.