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Your data matches 4 different statistics following compositions of up to 3 maps.
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Matching statistic: St000798
St000798: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 3
[1,2,3,4] => 0
[1,2,4,3] => 3
[1,3,2,4] => 2
[1,3,4,2] => 2
[1,4,2,3] => 3
[1,4,3,2] => 5
[2,1,3,4] => 1
[2,1,4,3] => 4
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 4
[3,1,2,4] => 2
[3,1,4,2] => 4
[3,2,1,4] => 3
[3,2,4,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 3
[4,1,2,3] => 3
[4,1,3,2] => 5
[4,2,1,3] => 4
[4,2,3,1] => 4
[4,3,1,2] => 5
[4,3,2,1] => 6
[1,2,3,4,5] => 0
[1,2,3,5,4] => 4
[1,2,4,3,5] => 3
[1,2,4,5,3] => 3
[1,2,5,3,4] => 4
[1,2,5,4,3] => 7
[1,3,2,4,5] => 2
[1,3,2,5,4] => 6
[1,3,4,2,5] => 2
[1,3,4,5,2] => 2
[1,3,5,2,4] => 3
[1,3,5,4,2] => 6
[1,4,2,3,5] => 3
[1,4,2,5,3] => 6
[1,4,3,2,5] => 5
[1,4,3,5,2] => 5
[1,4,5,2,3] => 3
[1,4,5,3,2] => 5
Description
The makl of a permutation.
According to [1], this is the sum of the number of occurrences of the vincular patterns $(1\underline{32})$, $(\underline{31}2)$, $(\underline{32}1)$ and $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000794
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000794: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
St000794: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 2
[3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,2,3] => 2
[1,4,2,3] => [1,3,4,2] => 3
[1,4,3,2] => [1,4,3,2] => 5
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [3,1,2,4] => 1
[2,3,4,1] => [4,1,2,3] => 1
[2,4,1,3] => [3,4,1,2] => 2
[2,4,3,1] => [4,3,1,2] => 4
[3,1,2,4] => [2,3,1,4] => 2
[3,1,4,2] => [4,2,3,1] => 4
[3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [4,1,3,2] => 3
[3,4,1,2] => [2,4,1,3] => 2
[3,4,2,1] => [4,2,1,3] => 3
[4,1,2,3] => [2,3,4,1] => 3
[4,1,3,2] => [3,2,4,1] => 5
[4,2,1,3] => [3,4,2,1] => 4
[4,2,3,1] => [3,1,4,2] => 4
[4,3,1,2] => [2,4,3,1] => 5
[4,3,2,1] => [4,3,2,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => 3
[1,2,4,5,3] => [1,2,5,3,4] => 3
[1,2,5,3,4] => [1,2,4,5,3] => 4
[1,2,5,4,3] => [1,2,5,4,3] => 7
[1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => 6
[1,3,4,2,5] => [1,4,2,3,5] => 2
[1,3,4,5,2] => [1,5,2,3,4] => 2
[1,3,5,2,4] => [1,4,5,2,3] => 3
[1,3,5,4,2] => [1,5,4,2,3] => 6
[1,4,2,3,5] => [1,3,4,2,5] => 3
[1,4,2,5,3] => [1,5,3,4,2] => 6
[1,4,3,2,5] => [1,4,3,2,5] => 5
[1,4,3,5,2] => [1,5,2,4,3] => 5
[1,4,5,2,3] => [1,3,5,2,4] => 3
[1,4,5,3,2] => [1,5,3,2,4] => 5
[1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 2
[1,3,4,5,2,7,6] => [1,5,2,3,4,7,6] => ? = 8
[1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 2
[1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 2
[1,3,4,5,7,2,6] => [1,6,7,2,3,4,5] => ? = 3
[1,3,4,5,7,6,2] => [1,7,6,2,3,4,5] => ? = 8
[1,3,4,6,2,5,7] => [1,5,6,2,3,4,7] => ? = 3
[1,3,4,6,2,7,5] => [1,7,5,6,2,3,4] => ? = 8
[1,3,4,6,5,2,7] => [1,6,5,2,3,4,7] => ? = 7
[1,3,4,6,5,7,2] => [1,7,2,3,4,6,5] => ? = 7
[1,3,4,6,7,2,5] => [1,5,7,2,3,4,6] => ? = 3
[1,3,4,6,7,5,2] => [1,7,5,2,3,4,6] => ? = 7
[1,3,4,7,2,5,6] => [1,5,6,7,2,3,4] => ? = 4
[1,3,4,7,2,6,5] => [1,6,5,7,2,3,4] => ? = 9
[1,3,4,7,5,2,6] => [1,6,7,5,2,3,4] => ? = 8
[1,3,4,7,5,6,2] => [1,6,2,3,4,7,5] => ? = 8
[1,3,4,7,6,2,5] => [1,5,7,6,2,3,4] => ? = 9
[1,3,4,7,6,5,2] => [1,7,6,5,2,3,4] => ? = 13
[1,3,5,2,6,4,7] => [1,6,4,5,2,3,7] => ? = 7
[1,3,5,2,6,7,4] => [1,7,4,5,2,3,6] => ? = 7
[1,3,5,2,7,4,6] => [1,6,7,4,5,2,3] => ? = 8
[1,3,5,2,7,6,4] => [1,7,6,4,5,2,3] => ? = 13
[1,3,5,4,2,6,7] => [1,5,4,2,3,6,7] => ? = 6
[1,3,5,4,2,7,6] => [1,5,4,2,3,7,6] => ? = 12
[1,3,5,4,6,2,7] => [1,6,2,3,5,4,7] => ? = 6
[1,3,5,4,6,7,2] => [1,7,2,3,5,4,6] => ? = 6
[1,3,5,4,7,2,6] => [1,6,7,2,3,5,4] => ? = 7
[1,3,5,4,7,6,2] => [1,7,6,2,3,5,4] => ? = 12
[1,3,5,6,2,7,4] => [1,7,4,6,2,3,5] => ? = 7
[1,3,5,6,4,2,7] => [1,6,4,2,3,5,7] => ? = 6
[1,3,5,6,4,7,2] => [1,7,2,3,6,4,5] => ? = 6
[1,3,5,6,7,2,4] => [1,4,7,2,3,5,6] => ? = 3
[1,3,5,6,7,4,2] => [1,7,4,2,3,5,6] => ? = 6
[1,3,5,7,2,4,6] => [1,4,6,7,2,3,5] => ? = 4
[1,3,5,7,2,6,4] => [1,6,4,7,2,3,5] => ? = 8
[1,3,5,7,4,2,6] => [1,6,7,4,2,3,5] => ? = 7
[1,3,5,7,4,6,2] => [1,6,2,3,7,4,5] => ? = 7
[1,3,5,7,6,2,4] => [1,4,7,6,2,3,5] => ? = 9
[1,3,5,7,6,4,2] => [1,7,6,4,2,3,5] => ? = 12
[1,3,6,2,4,7,5] => [1,4,7,5,6,2,3] => ? = 9
[1,3,6,2,5,4,7] => [1,5,4,6,2,3,7] => ? = 8
[1,3,6,2,5,7,4] => [1,7,4,5,6,2,3] => ? = 8
[1,3,6,2,7,4,5] => [1,5,7,4,6,2,3] => ? = 9
[1,3,6,2,7,5,4] => [1,7,5,4,6,2,3] => ? = 13
[1,3,6,4,2,5,7] => [1,5,6,4,2,3,7] => ? = 7
[1,3,6,4,2,7,5] => [1,7,5,6,4,2,3] => ? = 12
[1,3,6,4,5,2,7] => [1,5,2,3,6,4,7] => ? = 7
[1,3,6,4,5,7,2] => [1,7,2,3,5,6,4] => ? = 7
[1,3,6,4,7,2,5] => [1,5,7,2,3,6,4] => ? = 8
[1,3,6,4,7,5,2] => [1,7,5,2,3,6,4] => ? = 12
Description
The mak of a permutation.
According to [1], this is the sum of the number of occurrences of the vincular patterns $(2\underline{31})$, $(\underline{32}1)$, $(1\underline{32})$, $(\underline{21})$, where matches of the underlined letters must be adjacent.
Matching statistic: St000156
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000156: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 84%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000156: Permutations ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 84%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [2,3,1] => [3,2,1] => 2
[3,2,1] => [3,2,1] => [2,3,1] => 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 3
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 5
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 4
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 1
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 1
[2,4,1,3] => [3,4,1,2] => [4,1,3,2] => 2
[2,4,3,1] => [4,3,1,2] => [3,1,4,2] => 4
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,3,1] => [3,4,2,1] => 4
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 3
[3,2,4,1] => [4,1,3,2] => [3,4,1,2] => 3
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => 2
[3,4,2,1] => [4,2,1,3] => [2,4,1,3] => 3
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 3
[4,1,3,2] => [3,2,4,1] => [4,3,2,1] => 5
[4,2,1,3] => [3,4,2,1] => [2,4,3,1] => 4
[4,2,3,1] => [3,1,4,2] => [4,3,1,2] => 4
[4,3,1,2] => [2,4,3,1] => [3,2,4,1] => 5
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 3
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 3
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 7
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 6
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 2
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,2,4,3] => 3
[1,3,5,4,2] => [1,5,4,2,3] => [1,4,2,5,3] => 6
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => 3
[1,4,2,5,3] => [1,5,3,4,2] => [1,4,5,3,2] => 6
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 5
[1,4,3,5,2] => [1,5,2,4,3] => [1,4,5,2,3] => 5
[1,4,5,2,3] => [1,3,5,2,4] => [1,5,3,2,4] => 3
[1,4,5,3,2] => [1,5,3,2,4] => [1,3,5,2,4] => 5
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 6
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 5
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 5
[1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ? = 6
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => ? = 11
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 4
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 10
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => ? = 4
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => ? = 4
[1,2,3,5,7,4,6] => [1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => ? = 5
[1,2,3,5,7,6,4] => [1,2,3,7,6,4,5] => [1,2,3,6,4,7,5] => ? = 10
[1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 5
[1,2,3,6,4,7,5] => [1,2,3,7,5,6,4] => [1,2,3,6,7,5,4] => ? = 10
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => ? = 9
[1,2,3,6,5,7,4] => [1,2,3,7,4,6,5] => [1,2,3,6,7,4,5] => ? = 9
[1,2,3,6,7,4,5] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => ? = 5
[1,2,3,6,7,5,4] => [1,2,3,7,5,4,6] => [1,2,3,5,7,4,6] => ? = 9
[1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => ? = 6
[1,2,3,7,4,6,5] => [1,2,3,6,5,7,4] => [1,2,3,7,6,5,4] => ? = 11
[1,2,3,7,5,4,6] => [1,2,3,6,7,5,4] => [1,2,3,5,7,6,4] => ? = 10
[1,2,3,7,5,6,4] => [1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => ? = 10
[1,2,3,7,6,4,5] => [1,2,3,5,7,6,4] => [1,2,3,6,5,7,4] => ? = 11
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,5,6,7,4] => ? = 15
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 3
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 9
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 8
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => ? = 8
[1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => ? = 9
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => ? = 14
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => ? = 3
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => ? = 9
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => ? = 3
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => ? = 3
[1,2,4,5,7,3,6] => [1,2,6,7,3,4,5] => [1,2,7,3,4,6,5] => ? = 4
[1,2,4,5,7,6,3] => [1,2,7,6,3,4,5] => [1,2,6,3,4,7,5] => ? = 9
[1,2,4,6,3,5,7] => [1,2,5,6,3,4,7] => [1,2,6,3,5,4,7] => ? = 4
[1,2,4,6,3,7,5] => [1,2,7,5,6,3,4] => [1,2,6,3,7,5,4] => ? = 9
[1,2,4,6,5,3,7] => [1,2,6,5,3,4,7] => [1,2,5,3,6,4,7] => ? = 8
[1,2,4,6,5,7,3] => [1,2,7,3,4,6,5] => [1,2,6,3,7,4,5] => ? = 8
[1,2,4,6,7,3,5] => [1,2,5,7,3,4,6] => [1,2,7,3,5,4,6] => ? = 4
[1,2,4,6,7,5,3] => [1,2,7,5,3,4,6] => [1,2,5,3,7,4,6] => ? = 8
[1,2,4,7,3,5,6] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => ? = 5
[1,2,4,7,3,6,5] => [1,2,6,5,7,3,4] => [1,2,7,3,6,5,4] => ? = 10
[1,2,4,7,5,3,6] => [1,2,6,7,5,3,4] => [1,2,5,3,7,6,4] => ? = 9
[1,2,4,7,5,6,3] => [1,2,6,3,4,7,5] => [1,2,7,3,6,4,5] => ? = 9
[1,2,4,7,6,3,5] => [1,2,5,7,6,3,4] => [1,2,6,3,5,7,4] => ? = 10
[1,2,4,7,6,5,3] => [1,2,7,6,5,3,4] => [1,2,5,3,6,7,4] => ? = 14
[1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => ? = 4
[1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => ? = 10
Description
The Denert index of a permutation.
It is defined as
$$
\begin{align*}
den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\
&+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\
&+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \}
\end{align*}
$$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $exc$ is the number of weak exceedences, see [[St000155]].
Matching statistic: St001232
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 37%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 37%
Values
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 5
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 4
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? = 2
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 3
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 3
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 5
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 4
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 5
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 6
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 3
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 4
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 7
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 6
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? = 3
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 3
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 5
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 3
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[1,5,2,3,4] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 4
[1,5,2,4,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 7
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 7
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 9
[2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 5
[2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 4
[2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 4
[2,1,5,3,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 5
[2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 8
[2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 5
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3
[1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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