Your data matches 45 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000423
Mapping
St000423: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 4
[1,2,4,3] => 4
[1,3,2,4] => 3
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 3
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 1
[4,1,3,2] => 1
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 10
[1,2,3,5,4] => 10
[1,2,4,3,5] => 9
[1,2,4,5,3] => 9
[1,2,5,3,4] => 9
[1,2,5,4,3] => 9
[1,3,2,4,5] => 8
[1,3,2,5,4] => 8
[1,3,4,2,5] => 7
[1,3,4,5,2] => 7
[1,3,5,2,4] => 7
[1,3,5,4,2] => 7
[1,4,2,3,5] => 7
[1,4,2,5,3] => 7
[1,4,3,2,5] => 6
[1,4,3,5,2] => 6
[1,4,5,2,3] => 6
Description
The number of occurrences of the pattern 123 or of the pattern 132 in a permutation.
Matching statistic: St000428
(load all 2 compositions to match this statistic)
Mapping
St000428: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 1
[1,3,2] => 0
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 4
[1,2,4,3] => 2
[1,3,2,4] => 3
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 4
[2,1,4,3] => 2
[2,3,1,4] => 3
[2,3,4,1] => 1
[2,4,1,3] => 1
[2,4,3,1] => 0
[3,1,2,4] => 3
[3,1,4,2] => 1
[3,2,1,4] => 3
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 1
[4,1,3,2] => 0
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 10
[1,2,3,5,4] => 7
[1,2,4,3,5] => 8
[1,2,4,5,3] => 5
[1,2,5,3,4] => 5
[1,2,5,4,3] => 3
[1,3,2,4,5] => 9
[1,3,2,5,4] => 6
[1,3,4,2,5] => 7
[1,3,4,5,2] => 4
[1,3,5,2,4] => 4
[1,3,5,4,2] => 2
[1,4,2,3,5] => 7
[1,4,2,5,3] => 4
[1,4,3,2,5] => 6
[1,4,3,5,2] => 3
[1,4,5,2,3] => 2
Description
The number of occurrences of the pattern 123 or of the pattern 213 in a permutation.
Matching statistic: St000436
Mapping
St000436: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ? = 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 1
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 1
[1,4,2,3] => 0
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 3
[2,4,1,3] => 1
[2,4,3,1] => 3
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 4
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 3
[4,3,1,2] => 2
[4,3,2,1] => 4
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 1
[1,2,5,3,4] => 0
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 3
[1,3,5,2,4] => 1
[1,3,5,4,2] => 3
[1,4,2,3,5] => 0
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 3
[1,4,5,2,3] => 2
[1,4,5,3,2] => 4
Description
The number of occurrences of the pattern 231 or of the pattern 321 in a permutation.
Matching statistic: St000437
(load all 2 compositions to match this statistic)
Mapping
St000437: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ? = 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 1
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 1
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 4
[4,3,2,1] => 4
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 1
[1,2,5,4,3] => 1
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 1
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 1
[1,4,3,5,2] => 1
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
Description
The number of occurrences of the pattern 312 or of the pattern 321 in a permutation.
Matching statistic: St000456
(load all 5 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 43%
Values
[1] => [1] => ([],1)
 => ? = 0
[1,2] => [2] => ([],2)
 => ? ∊ {0,0}
[2,1] => [2] => ([],2)
 => ? ∊ {0,0}
[1,2,3] => [3] => ([],3)
 => ? ∊ {0,0,0,0,1}
[1,3,2] => [1,2] => ([(1,2)],3)
 => ? ∊ {0,0,0,0,1}
[2,1,3] => [3] => ([],3)
 => ? ∊ {0,0,0,0,1}
[2,3,1] => [3] => ([],3)
 => ? ∊ {0,0,0,0,1}
[3,1,2] => [3] => ([],3)
 => ? ∊ {0,0,0,0,1}
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,2,3,4] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[1,3,2,4] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[1,3,4,2] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[1,4,2,3] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[2,3,1,4] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[2,3,4,1] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[2,4,1,3] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,1,2,4] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[3,4,1,2] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => [4] => ([],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,3,3,3,3,4,4}
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,5] => [5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,5,2,3,4] => [1,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[2,1,3,4,5] => [5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,3,1,4,5] => [5] => ([],5)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[3,5,4,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,1,5,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
[5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[1,2,3,6,5,4] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[1,2,6,5,4,3] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6
[1,3,2,6,5,4] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000422
(load all 7 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 24%●distinct values known / distinct values provided: 14%
Values
[1] => [1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => [2] => ([],2)
 => 0
[2,1] => [1,2] => [2] => ([],2)
 => 0
[1,2,3] => [1,2,3] => [3] => ([],3)
 => 0
[1,3,2] => [1,2,3] => [3] => ([],3)
 => 0
[2,1,3] => [1,2,3] => [3] => ([],3)
 => 0
[2,3,1] => [1,2,3] => [3] => ([],3)
 => 0
[3,1,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {1,1}
[3,2,1] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => ? ∊ {1,1}
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
 => 0
[1,2,4,3] => [1,2,3,4] => [4] => ([],4)
 => 0
[1,3,2,4] => [1,2,3,4] => [4] => ([],4)
 => 0
[1,3,4,2] => [1,2,3,4] => [4] => ([],4)
 => 0
[1,4,2,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,4,3,2] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[2,1,3,4] => [1,2,3,4] => [4] => ([],4)
 => 0
[2,1,4,3] => [1,2,3,4] => [4] => ([],4)
 => 0
[2,3,1,4] => [1,2,3,4] => [4] => ([],4)
 => 0
[2,3,4,1] => [1,2,3,4] => [4] => ([],4)
 => 0
[2,4,1,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[2,4,3,1] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[3,1,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[3,1,4,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[3,2,1,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[3,2,4,1] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[3,4,1,2] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[3,4,2,1] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[4,1,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[4,1,3,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[4,2,1,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[4,2,3,1] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[4,3,1,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[4,3,2,1] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,3,3,3,3,4,4}
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,2,4,5] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,5,3,2] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[1,5,2,4,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[1,5,3,4,2] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,4,2,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,4,3,2] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,1,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[2,1,5,4,3] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[2,3,1,4,5] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => [5] => ([],5)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[2,3,5,4,1] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[2,4,1,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,1,5,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[2,4,3,1,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,3,5,1] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[2,4,5,1,3] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,5,3,1] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[2,5,1,4,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[2,5,3,4,1] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,5,4,1,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,5,4,3,1] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,2,5,4] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,4,2,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,4,5,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 6
[3,1,5,4,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,2,1,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,2,1,5,4] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,2,4,1,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,2,4,5,1] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[3,2,5,4,1] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,4,1,2,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,4,1,5,2] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,4,2,1,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,4,2,5,1] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,4,5,1,2] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,4,5,2,1] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,5,1,2,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,5,1,4,2] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000454
(load all 9 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 19% ●values known / values provided: 23%●distinct values known / distinct values provided: 19%
Values
[1] => [1] => [1] => ([],1)
 => 0
[1,2] => [1,2] => [1,2] => ([],2)
 => 0
[2,1] => [1,2] => [1,2] => ([],2)
 => 0
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[1,3,2] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[2,1,3] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
 => 0
[3,1,2] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? ∊ {1,1}
[3,2,1] => [1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => ? ∊ {1,1}
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[1,4,2,3] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[1,4,3,2] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
 => 0
[2,4,1,3] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[2,4,3,1] => [1,2,4,3] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[3,1,2,4] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[3,1,4,2] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[3,2,1,4] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[3,2,4,1] => [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[3,4,1,2] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[3,4,2,1] => [1,3,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[4,1,2,3] => [1,4,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[4,1,3,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[4,2,1,3] => [1,4,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,2,2,3,3,3,3,4,4}
[4,2,3,1] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[4,3,1,2] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[4,3,2,1] => [1,4,2,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,5,3,2] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,2,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,5,2,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,3,2,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,5,3,4,2] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,4,2,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,4,3,2] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[2,1,5,3,4] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,1,5,4,3] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => 0
[2,3,5,1,4] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
[2,4,1,3,5] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,1,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,3,1,5] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,3,5,1] => [1,2,4,5,3] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,5,1,3] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,5,3,1] => [1,2,4,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,5,1,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,5,1,4,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,5,3,1,4] => [1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,5,3,4,1] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,5,4,1,3] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,5,4,3,1] => [1,2,5,3,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,2,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,2,5,4] => [1,3,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,4,2,5] => [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,4,5,2] => [1,3,4,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,5,2,4] => [1,3,5,4,2] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,5,4,2] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,2,1,4,5] => [1,3,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,2,1,5,4] => [1,3,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,2,4,1,5] => [1,3,4,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,2,4,5,1] => [1,3,4,5,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,2,5,1,4] => [1,3,5,4,2] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,2,5,4,1] => [1,3,5,2,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,4,1,2,5] => [1,3,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,4,1,5,2] => [1,3,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,4,2,1,5] => [1,3,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,4,2,5,1] => [1,3,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[4,1,3,2,5] => [1,4,2,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
[4,1,5,2,3] => [1,4,2,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
[4,2,3,1,5] => [1,4,2,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000714
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 24%
Values
[1] => ([],1)
 => [1]
 => []
 => ? = 0
[1,2] => ([],2)
 => [1,1]
 => [1]
 => ? ∊ {0,0}
[2,1] => ([(0,1)],2)
 => [2]
 => []
 => ? ∊ {0,0}
[1,2,3] => ([],3)
 => [1,1,1]
 => [1,1]
 => 1
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1}
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1}
[2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,0,0,0,1}
[3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,0,0,0,1}
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,0,0,0,1}
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 3
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,2,2,3,3,3,4,4}
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,1,3,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 3
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 3
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 3
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 3
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 3
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 3
[1,2,3,4,5,6] => ([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,2,3,4,6,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,5,4,6] => ([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,3,5,6] => ([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => 0
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0
[1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
Description
The number of semistandard Young tableau of given shape, with entries at most 2. This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St000770
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 24%
Values
[1] => ([],1)
 => [1]
 => []
 => ? = 0
[1,2] => ([],2)
 => [1,1]
 => [1]
 => ? ∊ {0,0}
[2,1] => ([(0,1)],2)
 => [2]
 => []
 => ? ∊ {0,0}
[1,2,3] => ([],3)
 => [1,1,1]
 => [1,1]
 => 1
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1}
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1}
[2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,0,0,0,1}
[3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,0,0,0,1}
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,0,0,0,1}
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 4
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,1,3,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 4
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 4
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[1,2,3,4,5,6] => ([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[1,2,3,4,6,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,3,5,4,6] => ([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5,6] => ([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => 5
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
Description
The major index of an integer partition when read from bottom to top. This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top. For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
Matching statistic: St000815
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000815: Integer partitions ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 19%
Values
[1] => ([],1)
 => [1]
 => []
 => ? = 0
[1,2] => ([],2)
 => [1,1]
 => [1]
 => ? ∊ {0,0}
[2,1] => ([(0,1)],2)
 => [2]
 => []
 => ? ∊ {0,0}
[1,2,3] => ([],3)
 => [1,1,1]
 => [1,1]
 => 1
[1,3,2] => ([(1,2)],3)
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1}
[2,1,3] => ([(1,2)],3)
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,1}
[2,3,1] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,0,0,0,1}
[3,1,2] => ([(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,0,0,0,1}
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? ∊ {0,0,0,0,1}
[1,2,3,4] => ([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
[1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
[1,3,4,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[1,4,2,3] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
[2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
[2,3,1,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,1,2,4] => ([(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,2,3,3,3,3,4,4}
[1,2,3,4,5] => ([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,3,2,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,1,3,4,5] => ([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 3
[2,1,4,3,5] => ([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 3
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10}
[3,1,2,4,5] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
[1,2,3,4,5,6] => ([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[1,2,3,4,6,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,3,5,4,6] => ([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,3,5,6] => ([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
[1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => 4
[1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,4,6,5,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,3,4,6] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,5,6,4,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,6,3,4,5] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,6,3,5,4] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
[1,2,6,4,3,5] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
Description
The number of semistandard Young tableaux of partition weight of given shape. The weight of a semistandard Young tableaux is the sequence $(m_1, m_2,\dots)$, where $m_i$ is the number of occurrences of the number $i$ in the tableau. This statistic counts those tableaux whose weight is a weakly decreasing sequence. Alternatively, this is the sum of the entries in the column specified by the partition of the change of basis matrix from Schur functions to monomial symmetric functions.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000937The number of positive values of the symmetric group character corresponding to the partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000264The girth of a graph, which is not a tree. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001570The minimal number of edges to add to make a graph Hamiltonian.