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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000391
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%
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
[1,2] => 0 => [2] => 10 => 1
[2,1] => 1 => [1,1] => 11 => 3
[1,2,3] => 00 => [3] => 100 => 1
[1,3,2] => 01 => [2,1] => 101 => 4
[2,1,3] => 10 => [1,2] => 110 => 3
[2,3,1] => 01 => [2,1] => 101 => 4
[3,1,2] => 01 => [2,1] => 101 => 4
[3,2,1] => 11 => [1,1,1] => 111 => 6
[1,2,3,4] => 000 => [4] => 1000 => 1
[1,2,4,3] => 001 => [3,1] => 1001 => 5
[1,3,2,4] => 010 => [2,2] => 1010 => 4
[1,3,4,2] => 001 => [3,1] => 1001 => 5
[1,4,2,3] => 001 => [3,1] => 1001 => 5
[1,4,3,2] => 011 => [2,1,1] => 1011 => 8
[2,1,3,4] => 100 => [1,3] => 1100 => 3
[2,1,4,3] => 101 => [1,2,1] => 1101 => 7
[2,3,1,4] => 010 => [2,2] => 1010 => 4
[2,3,4,1] => 001 => [3,1] => 1001 => 5
[2,4,1,3] => 001 => [3,1] => 1001 => 5
[2,4,3,1] => 011 => [2,1,1] => 1011 => 8
[3,1,2,4] => 010 => [2,2] => 1010 => 4
[3,1,4,2] => 011 => [2,1,1] => 1011 => 8
[3,2,1,4] => 110 => [1,1,2] => 1110 => 6
[3,2,4,1] => 011 => [2,1,1] => 1011 => 8
[3,4,1,2] => 001 => [3,1] => 1001 => 5
[3,4,2,1] => 101 => [1,2,1] => 1101 => 7
[4,1,2,3] => 001 => [3,1] => 1001 => 5
[4,1,3,2] => 011 => [2,1,1] => 1011 => 8
[4,2,1,3] => 101 => [1,2,1] => 1101 => 7
[4,2,3,1] => 011 => [2,1,1] => 1011 => 8
[4,3,1,2] => 011 => [2,1,1] => 1011 => 8
[4,3,2,1] => 111 => [1,1,1,1] => 1111 => 10
[1,2,3,4,5] => 0000 => [5] => 10000 => 1
[1,2,3,5,4] => 0001 => [4,1] => 10001 => 6
[1,2,4,3,5] => 0010 => [3,2] => 10010 => 5
[1,2,4,5,3] => 0001 => [4,1] => 10001 => 6
[1,2,5,3,4] => 0001 => [4,1] => 10001 => 6
[1,2,5,4,3] => 0011 => [3,1,1] => 10011 => 10
[1,3,2,4,5] => 0100 => [2,3] => 10100 => 4
[1,3,2,5,4] => 0101 => [2,2,1] => 10101 => 9
[1,3,4,2,5] => 0010 => [3,2] => 10010 => 5
[1,3,4,5,2] => 0001 => [4,1] => 10001 => 6
[1,3,5,2,4] => 0001 => [4,1] => 10001 => 6
[1,3,5,4,2] => 0011 => [3,1,1] => 10011 => 10
[1,4,2,3,5] => 0010 => [3,2] => 10010 => 5
[1,4,2,5,3] => 0011 => [3,1,1] => 10011 => 10
[1,4,3,2,5] => 0110 => [2,1,2] => 10110 => 8
[1,4,3,5,2] => 0011 => [3,1,1] => 10011 => 10
[1,4,5,2,3] => 0001 => [4,1] => 10001 => 6
Description
The sum of the positions of the ones in a binary word.
Matching statistic: St000471
St000471: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[1] => ? = 1 - 1
[1,2] => 2 = 3 - 1
[2,1] => 0 = 1 - 1
[1,2,3] => 5 = 6 - 1
[1,3,2] => 3 = 4 - 1
[2,1,3] => 3 = 4 - 1
[2,3,1] => 3 = 4 - 1
[3,1,2] => 2 = 3 - 1
[3,2,1] => 0 = 1 - 1
[1,2,3,4] => 9 = 10 - 1
[1,2,4,3] => 6 = 7 - 1
[1,3,2,4] => 7 = 8 - 1
[1,3,4,2] => 7 = 8 - 1
[1,4,2,3] => 7 = 8 - 1
[1,4,3,2] => 4 = 5 - 1
[2,1,3,4] => 7 = 8 - 1
[2,1,4,3] => 4 = 5 - 1
[2,3,1,4] => 7 = 8 - 1
[2,3,4,1] => 7 = 8 - 1
[2,4,1,3] => 7 = 8 - 1
[2,4,3,1] => 4 = 5 - 1
[3,1,2,4] => 6 = 7 - 1
[3,1,4,2] => 4 = 5 - 1
[3,2,1,4] => 4 = 5 - 1
[3,2,4,1] => 4 = 5 - 1
[3,4,1,2] => 6 = 7 - 1
[3,4,2,1] => 4 = 5 - 1
[4,1,2,3] => 5 = 6 - 1
[4,1,3,2] => 3 = 4 - 1
[4,2,1,3] => 3 = 4 - 1
[4,2,3,1] => 3 = 4 - 1
[4,3,1,2] => 2 = 3 - 1
[4,3,2,1] => 0 = 1 - 1
[1,2,3,4,5] => 14 = 15 - 1
[1,2,3,5,4] => 10 = 11 - 1
[1,2,4,3,5] => 11 = 12 - 1
[1,2,4,5,3] => 11 = 12 - 1
[1,2,5,3,4] => 11 = 12 - 1
[1,2,5,4,3] => 7 = 8 - 1
[1,3,2,4,5] => 12 = 13 - 1
[1,3,2,5,4] => 8 = 9 - 1
[1,3,4,2,5] => 12 = 13 - 1
[1,3,4,5,2] => 12 = 13 - 1
[1,3,5,2,4] => 12 = 13 - 1
[1,3,5,4,2] => 8 = 9 - 1
[1,4,2,3,5] => 12 = 13 - 1
[1,4,2,5,3] => 9 = 10 - 1
[1,4,3,2,5] => 9 = 10 - 1
[1,4,3,5,2] => 9 = 10 - 1
[1,4,5,2,3] => 12 = 13 - 1
[1,4,5,3,2] => 9 = 10 - 1
[1,4,6,5,7,2,3] => ? ∊ 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- 1
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- 1
[1,5,2,4,6,3,7] => ? ∊ 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- 1
[1,5,2,6,4,3,7] => ? ∊ 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- 1
Description
The sum of the ascent tops of a permutation.
Matching statistic: St000111
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000111: Permutations ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 73%
Values
[1] => 0 = 1 - 1
[1,2] => 0 = 1 - 1
[2,1] => 2 = 3 - 1
[1,2,3] => 0 = 1 - 1
[1,3,2] => 3 = 4 - 1
[2,1,3] => 2 = 3 - 1
[2,3,1] => 3 = 4 - 1
[3,1,2] => 3 = 4 - 1
[3,2,1] => 5 = 6 - 1
[1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => 4 = 5 - 1
[1,3,2,4] => 3 = 4 - 1
[1,3,4,2] => 4 = 5 - 1
[1,4,2,3] => 4 = 5 - 1
[1,4,3,2] => 7 = 8 - 1
[2,1,3,4] => 2 = 3 - 1
[2,1,4,3] => 6 = 7 - 1
[2,3,1,4] => 3 = 4 - 1
[2,3,4,1] => 4 = 5 - 1
[2,4,1,3] => 4 = 5 - 1
[2,4,3,1] => 7 = 8 - 1
[3,1,2,4] => 3 = 4 - 1
[3,1,4,2] => 7 = 8 - 1
[3,2,1,4] => 5 = 6 - 1
[3,2,4,1] => 7 = 8 - 1
[3,4,1,2] => 4 = 5 - 1
[3,4,2,1] => 6 = 7 - 1
[4,1,2,3] => 4 = 5 - 1
[4,1,3,2] => 7 = 8 - 1
[4,2,1,3] => 6 = 7 - 1
[4,2,3,1] => 7 = 8 - 1
[4,3,1,2] => 7 = 8 - 1
[4,3,2,1] => 9 = 10 - 1
[1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => 5 = 6 - 1
[1,2,4,3,5] => 4 = 5 - 1
[1,2,4,5,3] => 5 = 6 - 1
[1,2,5,3,4] => 5 = 6 - 1
[1,2,5,4,3] => 9 = 10 - 1
[1,3,2,4,5] => 3 = 4 - 1
[1,3,2,5,4] => 8 = 9 - 1
[1,3,4,2,5] => 4 = 5 - 1
[1,3,4,5,2] => 5 = 6 - 1
[1,3,5,2,4] => 5 = 6 - 1
[1,3,5,4,2] => 9 = 10 - 1
[1,4,2,3,5] => 4 = 5 - 1
[1,4,2,5,3] => 9 = 10 - 1
[1,4,3,2,5] => 7 = 8 - 1
[1,4,3,5,2] => 9 = 10 - 1
[1,4,5,2,3] => 5 = 6 - 1
[1,2,3,4,5,6,7] => ? ∊ 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- 1
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- 1
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- 1
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- 1
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- 1
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- 1
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- 1
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- 1
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- 1
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- 1
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19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,19,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,20,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,21,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,22,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,23,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,25,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,26,28} - 1
Description
The sum of the descent tops (or Genocchi descents) of a permutation.
This statistic is given by
π↦∑i∈Des(π)πi.
Matching statistic: St000699
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> ? = 1
[1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,3}
[2,1] => ([],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,3}
[1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,6}
[1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,6}
[2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,6}
[2,3,1] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,6}
[3,1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,6}
[3,2,1] => ([],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,3,4,4,4,6}
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,4,2,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,1,3,2] => ([(1,2),(1,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,2,1,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,2,3,1] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,3,1,2] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,3,2,1] => ([],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,4,6,2,5] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,5,2,6,4] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,5,4,2,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,3,6,4,2,5] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,2,3,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,2,6,3,5] => ([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,5,2,6,3] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,5,3,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,5,3,6,2] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,4,6,2,3,5] => ([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,2,3,6,4] => ([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,2,4,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,2,4,6,3] => ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,3,2,4,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,3,4,2,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,3,4,6,2] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,5,4,2,3,6] => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,6,3,4,2,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[1,6,4,2,3,5] => ([(0,2),(0,3),(0,4),(1,5),(3,5),(4,1)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,1,5,3,4,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,3,5,1,6,4] => ([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,1,5,3,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,1,5,6,3] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,1,6,3,5] => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,1,6,5,3] => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,5,1,3,6] => ([(0,4),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,5,3,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,5,3,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,4,6,1,5,3] => ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,1,6,3,4] => ([(0,4),(0,5),(1,2),(1,4),(2,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,1,6,4,3] => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,3,1,4,6] => ([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,3,4,1,6] => ([(0,5),(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,3,4,6,1] => ([(1,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,5,4,1,6,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,5),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[2,6,4,1,5,3] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[3,1,4,5,2,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
[3,1,5,2,4,6] => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 12
Description
The toughness times the least common multiple of 1,...,n-1 of a non-complete graph.
A graph G is t-tough if G cannot be split into k different connected components by the removal of fewer than tk vertices for all integers k>1.
The toughness of G is the maximal number t such that G is t-tough. It is a rational number except for the complete graph, where it is infinity. The toughness of a disconnected graph is zero.
This statistic is the toughness multiplied by the least common multiple of 1,…,n−1, where n is the number of vertices of G.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 35%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 35%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {4,4,4,6}
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,4,6}
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,6}
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,4,6}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,8,8,10}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 7
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> 9
[1,2,3,5,6,4] => [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> 10
[1,2,4,5,3,6] => [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> 7
[1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> 4
[1,2,4,6,3,5] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 9
[1,3,4,2,5,6] => [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> 7
[1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> 5
[1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> 3
[1,3,4,6,2,5] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
[1,3,4,6,5,2] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
[1,3,5,2,4,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> 8
[1,3,5,2,6,4] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> 7
[1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> 8
[1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> 7
[1,3,5,6,2,4] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,3,5,6,4,2] => [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001645
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 15%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {1,3}
[2,1] => [1,2] => [1,2] => ([],2)
=> ? ∊ {1,3}
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,3,4,6}
[1,3,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 4
[2,1,3] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 4
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,3,4,6}
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,3,4,6}
[3,2,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,3,4,6}
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[1,2,4,3] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[1,3,2,4] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[1,3,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 8
[1,4,2,3] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[1,4,3,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[2,1,3,4] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 8
[2,1,4,3] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[2,3,1,4] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[2,4,1,3] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[3,1,2,4] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[3,1,4,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[3,2,1,4] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[3,2,4,1] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[4,1,3,2] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[4,2,1,3] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,7,7,7,8,8,8,8,8,10}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,4,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,5,3,4] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,2,5,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,2,5,4] => [1,3,2,5,4] => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,5,2,4] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,3,5,4,2] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,2,3,5] => [1,4,2,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,2,5,3] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,3,2,5] => [1,4,2,5,3] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,3,5,2] => [1,4,2,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,4,5,3,2] => [1,4,5,2,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,2,3,4] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,2,4,3] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,3,2,4] => [1,5,2,4,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,3,4,2] => [1,5,2,3,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {1,3,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,13,13,13,13,13,13,13,15}
[1,5,2,6,3,7,4] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[1,5,2,6,4,3,7] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[1,5,3,7,2,6,4] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[1,5,3,7,4,2,6] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[1,5,4,2,6,3,7] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[1,5,4,3,7,2,6] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[2,6,1,5,3,7,4] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[2,6,1,5,4,3,7] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[2,6,3,7,1,5,4] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[2,6,3,7,4,1,5] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[2,6,4,1,5,3,7] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[2,6,4,3,7,1,5] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[3,7,1,5,2,6,4] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[3,7,1,5,4,2,6] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[3,7,2,6,1,5,4] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[3,7,2,6,4,1,5] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[3,7,4,1,5,2,6] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[3,7,4,2,6,1,5] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[4,1,5,2,6,3,7] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[4,1,5,3,7,2,6] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[4,2,6,1,5,3,7] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[4,2,6,3,7,1,5] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[4,3,7,1,5,2,6] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
[4,3,7,2,6,1,5] => [1,5,2,6,3,7,4] => [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 7
Description
The pebbling number of a connected graph.
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