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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000058
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [2,1] => [1,2] => [1,2] => 1
[2,1] => [1,2] => [1,2] => [1,2] => 1
[1,2,3] => [3,2,1] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [2,3,1] => [1,2,3] => [1,2,3] => 1
[2,1,3] => [3,1,2] => [1,2,3] => [1,2,3] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 2
[3,1,2] => [2,1,3] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 2
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [1,4,3,2] => 2
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [1,4,3,2] => 2
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [1,4,3,2] => 2
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 2
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 2
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [1,4,3,2] => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [1,4,3,2] => 2
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [1,4,3,2] => 2
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 2
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [1,2,5,4,3] => 2
Description
The order of a permutation.
$\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Matching statistic: St000392
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => => ? = 1 - 1
[1,2] => [2,1] => [1,2] => 0 => 0 = 1 - 1
[2,1] => [1,2] => [1,2] => 0 => 0 = 1 - 1
[1,2,3] => [3,2,1] => [1,2,3] => 00 => 0 = 1 - 1
[1,3,2] => [3,1,2] => [1,2,3] => 00 => 0 = 1 - 1
[2,1,3] => [2,3,1] => [1,2,3] => 00 => 0 = 1 - 1
[2,3,1] => [2,1,3] => [1,3,2] => 01 => 1 = 2 - 1
[3,1,2] => [1,3,2] => [1,3,2] => 01 => 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,2,4,3] => [4,3,1,2] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,3,4,2] => [4,2,1,3] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,4,2,3] => [4,1,3,2] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,4,3,2] => [4,1,2,3] => [1,2,3,4] => 000 => 0 = 1 - 1
[2,1,3,4] => [3,4,2,1] => [1,2,3,4] => 000 => 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => 000 => 0 = 1 - 1
[2,3,1,4] => [3,2,4,1] => [1,2,4,3] => 001 => 1 = 2 - 1
[2,3,4,1] => [3,2,1,4] => [1,4,2,3] => 010 => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => 010 => 1 = 2 - 1
[2,4,3,1] => [3,1,2,4] => [1,2,4,3] => 001 => 1 = 2 - 1
[3,1,2,4] => [2,4,3,1] => [1,2,4,3] => 001 => 1 = 2 - 1
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => 010 => 1 = 2 - 1
[3,2,1,4] => [2,3,4,1] => [1,2,3,4] => 000 => 0 = 1 - 1
[3,2,4,1] => [2,3,1,4] => [1,4,2,3] => 010 => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => 010 => 1 = 2 - 1
[3,4,2,1] => [2,1,3,4] => [1,3,4,2] => 001 => 1 = 2 - 1
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => 010 => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 010 => 1 = 2 - 1
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => 001 => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => 001 => 1 = 2 - 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,4,2,3,5] => 0100 => 1 = 2 - 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,4,2,3,5] => 0100 => 1 = 2 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,4,2,3,5] => 0100 => 1 = 2 - 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,4,2,3,5] => 0100 => 1 = 2 - 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,3,4,2,5] => 0010 => 1 = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000264
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 64%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1] => ([],1)
=> ? = 1 + 1
[1,2] => [1,2] => [1,2] => ([],2)
=> ? = 1 + 1
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> ? = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? = 1 + 1
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1 + 1
[2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? = 1 + 1
[2,3,1] => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
[3,1,2] => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
[3,2,1] => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 1 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 1 + 1
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[2,3,4,1] => [4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,1,3] => [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,4,3,1] => [4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
[3,1,4,2] => [4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[3,2,4,1] => [4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,4,1,2] => [4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[3,4,2,1] => [4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,1,2,3] => [4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,1,3,2] => [4,2,3,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[4,2,1,3] => [4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,2,3,1] => [4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,3,1,2] => [4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 2 + 1
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? = 1 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ? = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,2,3,4] => [1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,2,4,3] => [1,5,3,4,2] => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,3,4,1,5] => [4,2,3,1,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,4,5,1] => [5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,5,1,4] => [4,2,5,1,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,3,5,4,1] => [5,2,4,3,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,1,3,5] => [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,1,5,3] => [3,5,1,4,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,3,5,1] => [5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,4,5,1,3] => [4,5,3,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,1,3,4] => [3,5,1,4,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,1,4,3] => [3,5,1,4,2] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,3,1,4] => [4,5,3,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,5,4,1,3] => [4,5,3,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,4,2,5] => [4,2,3,1,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,4,5,2] => [5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,5,2,4] => [4,2,5,1,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,1,5,4,2] => [5,2,4,3,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2,4,1,5] => [4,2,3,1,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2,4,5,1] => [5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2,5,1,4] => [4,2,5,1,3] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,2,5,4,1] => [5,2,4,3,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,4,1,5,2] => [5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,4,2,5,1] => [5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,5,1,2,4] => [4,5,3,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[3,5,2,1,4] => [4,5,3,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,2,3,5] => [4,2,3,1,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,2,5,3] => [5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,3,2,5] => [4,2,3,1,5] => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,3,5,2] => [5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,5,2,3] => [5,2,4,3,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,1,5,3,2] => [5,2,4,3,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,2,1,5,3] => [5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,2,3,5,1] => [5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,3,1,5,2] => [5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,3,2,5,1] => [5,3,2,4,1] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,1,2,3,4] => [5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,1,2,4,3] => [5,2,3,4,1] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,1,3,2,4] => [5,2,4,3,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[5,1,3,4,2] => [5,2,4,3,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000298
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2)
=> 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,3,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,1,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[3,2,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [1,4,3,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,4,3,2] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,3,1,4] => [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [4,2,3,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[2,4,1,3] => [4,2,1,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2
[2,4,3,1] => [3,2,4,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,2,4,1] => [4,3,2,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,4,1,2] => [4,1,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[4,1,2,3] => [4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 2
[4,1,3,2] => [4,3,1,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,1,3] => [2,4,1,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[4,2,3,1] => [3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [3,1,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,3,5,2,4] => [1,5,3,2,4] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
=> 2
[1,3,5,4,2] => [1,4,3,5,2] => [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,4,2,5,3] => [1,4,5,2,3] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,4,5,2,3] => [1,5,2,4,3] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,2,3,6,4,7,5] => [1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,2,3,7,5,6,4] => [1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,2,4,6,7,5,3] => [1,2,5,4,7,6,3] => [1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 2
[1,2,5,3,6,7,4] => [1,2,5,7,3,6,4] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 2
[1,2,5,3,7,4,6] => [1,2,5,7,3,4,6] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 2
[1,2,5,6,3,7,4] => [1,2,6,7,5,3,4] => [1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 2
[1,2,5,6,4,3,7] => [1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,2,5,7,4,6,3] => [1,2,6,7,5,4,3] => [1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 2
[1,2,6,3,7,4,5] => [1,2,6,7,4,3,5] => [1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 2
[1,2,6,4,3,5,7] => [1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,2,6,4,3,7,5] => [1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,2,6,4,7,5,3] => [1,2,5,7,6,3,4] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 2
[1,2,6,7,4,5,3] => [1,2,5,7,4,6,3] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 2
[1,2,6,7,5,3,4] => [1,2,5,3,7,6,4] => [1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 2
[1,2,7,4,5,3,6] => [1,2,5,7,4,3,6] => [1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ? = 2
[1,2,7,4,6,3,5] => [1,2,6,7,3,4,5] => [1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 2
[1,2,7,5,3,6,4] => [1,2,5,7,6,4,3] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7)
=> ? = 2
[1,2,7,5,6,4,3] => [1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,3,2,5,6,4,7] => [1,3,2,6,5,4,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,3,2,6,4,5,7] => [1,3,2,6,4,5,7] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,3,2,6,4,7,5] => [1,3,2,6,7,4,5] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,3,2,7,5,6,4] => [1,3,2,6,7,5,4] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? = 2
[1,3,4,2,5,6,7] => [1,4,3,2,5,6,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,3,4,2,5,7,6] => [1,4,3,2,5,7,6] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,3,4,2,6,5,7] => [1,4,3,2,6,5,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,3,4,2,6,7,5] => [1,4,3,2,7,6,5] => [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 2
[1,3,4,2,7,5,6] => [1,4,3,2,7,5,6] => [1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ? = 2
[1,3,4,2,7,6,5] => [1,4,3,2,6,7,5] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,3,4,6,7,2,5] => [1,7,3,4,2,6,5] => [1,2,7,5,3,4,6] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 2
[1,3,4,6,7,5,2] => [1,5,3,4,7,6,2] => [1,2,5,7,3,4,6] => ([(0,5),(2,6),(3,2),(4,1),(4,6),(5,3),(5,4)],7)
=> ? = 2
[1,3,5,2,6,7,4] => [1,5,3,7,2,6,4] => [1,2,5,3,4,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
=> ? = 2
[1,3,5,2,7,4,6] => [1,5,3,7,2,4,6] => [1,2,5,3,4,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
=> ? = 2
[1,3,5,6,2,7,4] => [1,6,3,7,5,2,4] => [1,2,6,3,4,7,5] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> ? = 2
[1,3,5,6,4,2,7] => [1,4,3,6,5,2,7] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[1,3,5,6,7,2,4] => [1,7,3,2,5,6,4] => [1,2,7,4,3,5,6] => ([(0,5),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
[1,3,5,6,7,4,2] => [1,4,3,7,5,6,2] => [1,2,4,7,3,5,6] => ([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
=> ? = 2
[1,3,5,7,2,6,4] => [1,7,3,6,5,2,4] => [1,2,7,4,6,3,5] => ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
=> ? = 2
[1,3,5,7,4,2,6] => [1,4,3,7,5,2,6] => [1,2,4,7,6,3,5] => ([(0,5),(3,6),(4,1),(4,2),(4,6),(5,3),(5,4)],7)
=> ? = 2
[1,3,5,7,4,6,2] => [1,6,3,7,5,4,2] => [1,2,6,4,7,3,5] => ([(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2),(4,3)],7)
=> ? = 2
[1,3,5,7,6,2,4] => [1,6,3,2,5,7,4] => [1,2,6,7,4,3,5] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(5,4)],7)
=> ? = 2
[1,3,5,7,6,4,2] => [1,4,3,6,5,7,2] => [1,2,4,6,7,3,5] => ([(0,5),(2,6),(3,1),(4,3),(4,6),(5,2),(5,4)],7)
=> ? = 2
[1,3,6,2,5,7,4] => [1,6,3,7,2,5,4] => [1,2,6,5,3,4,7] => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 2
[1,3,6,2,7,4,5] => [1,6,3,7,4,2,5] => [1,2,6,3,4,7,5] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
=> ? = 2
[1,3,6,4,2,5,7] => [1,4,3,6,2,5,7] => [1,2,4,6,5,3,7] => ([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
=> ? = 2
[1,3,6,4,2,7,5] => [1,4,3,6,7,2,5] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 2
[1,3,6,4,7,2,5] => [1,7,3,6,2,4,5] => [1,2,7,5,3,4,6] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
=> ? = 2
[1,3,6,4,7,5,2] => [1,5,3,7,6,2,4] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> ? = 2
[1,3,6,5,7,2,4] => [1,7,3,2,6,5,4] => [1,2,7,4,3,5,6] => ([(0,5),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ? = 2
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000485
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => ? = 1
[1,2] => [1,2] => [1,2] => [1,2] => 1
[2,1] => [2,1] => [1,2] => [1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [1,2,3] => [1,2,3] => 1
[2,1,3] => [2,1,3] => [1,2,3] => [1,2,3] => 1
[2,3,1] => [3,2,1] => [1,3,2] => [1,3,2] => 2
[3,1,2] => [3,1,2] => [1,3,2] => [1,3,2] => 2
[3,2,1] => [2,3,1] => [1,2,3] => [1,2,3] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 2
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,4,3,2] => [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2
[2,3,4,1] => [4,2,3,1] => [1,4,2,3] => [1,4,3,2] => 2
[2,4,1,3] => [4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 2
[2,4,3,1] => [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 2
[3,1,2,4] => [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[3,1,4,2] => [4,3,1,2] => [1,4,2,3] => [1,4,3,2] => 2
[3,2,1,4] => [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 1
[3,2,4,1] => [4,3,2,1] => [1,4,2,3] => [1,4,3,2] => 2
[3,4,1,2] => [4,1,3,2] => [1,4,2,3] => [1,4,3,2] => 2
[3,4,2,1] => [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 2
[4,1,2,3] => [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[4,1,3,2] => [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2
[4,2,1,3] => [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 2
[4,2,3,1] => [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 2
[4,3,1,2] => [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 2
[4,3,2,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,3,4,5,2] => [1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,3,5,2,4] => [1,5,3,2,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,5,4,2] => [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,4,2,5,3] => [1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,4,3,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,4,5,2,3] => [1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => 2
[1,4,5,3,2] => [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 2
[2,3,4,5,7,1,6] => [7,2,3,4,5,1,6] => [1,7,6,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 2
[2,3,4,5,7,6,1] => [6,2,3,4,5,7,1] => [1,6,7,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 2
[2,3,4,6,5,7,1] => [7,2,3,4,6,5,1] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 2
[2,3,4,6,7,1,5] => [7,2,3,4,1,6,5] => [1,7,5,2,3,4,6] => [1,7,6,4,5,3,2] => ? = 2
[2,3,4,6,7,5,1] => [5,2,3,4,7,6,1] => [1,5,7,2,3,4,6] => [1,6,7,4,5,2,3] => ? = 2
[2,3,4,7,1,5,6] => [7,2,3,4,1,5,6] => [1,7,6,5,2,3,4] => [1,7,6,5,4,3,2] => ? = 2
[2,3,4,7,1,6,5] => [6,2,3,4,7,1,5] => [1,6,2,3,4,5,7] => [1,6,3,4,5,2,7] => ? = 2
[2,3,4,7,5,1,6] => [5,2,3,4,7,1,6] => [1,5,7,6,2,3,4] => [1,7,6,5,4,3,2] => ? = 2
[2,3,4,7,6,1,5] => [6,2,3,4,1,7,5] => [1,6,7,5,2,3,4] => [1,7,6,5,4,3,2] => ? = 2
[2,3,4,7,6,5,1] => [5,2,3,4,6,7,1] => [1,5,6,7,2,3,4] => [1,7,6,5,4,3,2] => ? = 2
[2,3,5,1,6,7,4] => [7,2,3,5,1,6,4] => [1,7,4,5,2,3,6] => [1,7,6,5,4,3,2] => ? = 2
[2,3,5,1,7,4,6] => [7,2,3,5,1,4,6] => [1,7,6,4,5,2,3] => [1,7,6,5,4,3,2] => ? = 2
[2,3,5,4,6,7,1] => [7,2,3,5,4,6,1] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 2
[2,3,5,4,7,1,6] => [7,2,3,5,4,1,6] => [1,7,6,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 2
[2,3,5,4,7,6,1] => [6,2,3,5,4,7,1] => [1,6,7,2,3,4,5] => [1,7,6,4,5,3,2] => ? = 2
[2,3,5,6,1,7,4] => [7,2,3,6,5,1,4] => [1,7,4,6,2,3,5] => [1,7,6,5,4,3,2] => ? = 2
[2,3,5,6,4,7,1] => [7,2,3,6,5,4,1] => [1,7,2,3,4,6,5] => [1,7,3,4,5,6,2] => ? = 2
[2,3,5,6,7,1,4] => [7,2,3,1,5,6,4] => [1,7,4,2,3,5,6] => [1,7,5,4,3,6,2] => ? = 2
[2,3,5,6,7,4,1] => [4,2,3,7,5,6,1] => [1,4,7,2,3,5,6] => [1,5,7,4,2,6,3] => ? = 2
[2,3,5,7,1,4,6] => [7,2,3,1,5,4,6] => [1,7,6,4,2,3,5] => [1,7,6,5,4,3,2] => ? = 2
[2,3,5,7,1,6,4] => [6,2,3,7,5,1,4] => [1,6,2,3,4,7,5] => [1,7,3,4,5,6,2] => ? = 2
[2,3,5,7,4,1,6] => [4,2,3,7,5,1,6] => [1,4,7,6,2,3,5] => [1,6,7,5,4,2,3] => ? = 2
[2,3,5,7,4,6,1] => [6,2,3,7,5,4,1] => [1,6,4,7,2,3,5] => [1,7,6,5,4,3,2] => ? = 2
[2,3,5,7,6,1,4] => [6,2,3,1,5,7,4] => [1,6,7,4,2,3,5] => [1,7,6,5,4,3,2] => ? = 2
[2,3,5,7,6,4,1] => [4,2,3,6,5,7,1] => [1,4,6,7,2,3,5] => [1,6,7,5,4,2,3] => ? = 2
[2,3,6,1,5,7,4] => [7,2,3,6,1,5,4] => [1,7,4,6,5,2,3] => [1,7,6,5,4,3,2] => ? = 2
[2,3,6,1,7,4,5] => [7,2,3,6,1,4,5] => [1,7,5,2,3,4,6] => [1,7,6,4,5,3,2] => ? = 2
[2,3,6,4,5,7,1] => [7,2,3,6,4,5,1] => [1,7,2,3,4,6,5] => [1,7,3,4,5,6,2] => ? = 2
[2,3,6,4,7,1,5] => [7,2,3,6,4,1,5] => [1,7,5,4,6,2,3] => [1,7,6,5,4,3,2] => ? = 2
[2,3,6,4,7,5,1] => [5,2,3,7,6,4,1] => [1,5,6,4,7,2,3] => [1,7,6,4,5,3,2] => ? = 2
[2,3,6,5,4,7,1] => [7,2,3,5,6,4,1] => [1,7,2,3,4,5,6] => [1,7,3,4,5,6,2] => ? = 2
[2,3,6,5,7,1,4] => [7,2,3,1,6,5,4] => [1,7,4,2,3,5,6] => [1,7,5,4,3,6,2] => ? = 2
[2,3,6,5,7,4,1] => [4,2,3,7,6,5,1] => [1,4,7,2,3,5,6] => [1,5,7,4,2,6,3] => ? = 2
[2,3,6,7,1,4,5] => [7,2,3,1,4,6,5] => [1,7,5,4,2,3,6] => [1,7,6,5,4,3,2] => ? = 2
[2,3,6,7,1,5,4] => [5,2,3,7,1,6,4] => [1,5,2,3,4,7,6] => [1,5,3,4,2,7,6] => ? = 2
[2,3,6,7,4,1,5] => [4,2,3,7,1,6,5] => [1,4,7,5,2,3,6] => [1,6,7,5,4,2,3] => ? = 2
[2,3,6,7,4,5,1] => [5,2,3,7,4,6,1] => [1,5,4,7,2,3,6] => [1,6,5,7,3,2,4] => ? = 2
[2,3,6,7,5,1,4] => [5,2,3,1,7,6,4] => [1,5,7,4,2,3,6] => [1,6,7,5,4,2,3] => ? = 2
[2,3,7,1,4,5,6] => [7,2,3,1,4,5,6] => [1,7,6,5,4,2,3] => [1,7,6,5,4,3,2] => ? = 2
[2,3,7,1,4,6,5] => [6,2,3,1,7,4,5] => [1,6,4,2,3,5,7] => [1,6,5,4,3,2,7] => ? = 2
[2,3,7,1,5,4,6] => [5,2,3,7,1,4,6] => [1,5,2,3,4,7,6] => [1,5,3,4,2,7,6] => ? = 2
[2,3,7,1,5,6,4] => [6,2,3,7,1,5,4] => [1,6,5,2,3,4,7] => [1,6,5,4,3,2,7] => ? = 2
[2,3,7,1,6,4,5] => [6,2,3,7,1,4,5] => [1,6,4,7,5,2,3] => [1,7,6,5,4,3,2] => ? = 2
[2,3,7,1,6,5,4] => [5,2,3,6,7,1,4] => [1,5,7,4,6,2,3] => [1,7,6,5,4,3,2] => ? = 2
[2,3,7,4,1,5,6] => [4,2,3,7,1,5,6] => [1,4,7,6,5,2,3] => [1,7,6,5,4,3,2] => ? = 2
[2,3,7,4,1,6,5] => [4,2,3,6,7,1,5] => [1,4,6,2,3,5,7] => [1,5,6,4,2,3,7] => ? = 2
[2,3,7,4,5,1,6] => [5,2,3,7,4,1,6] => [1,5,4,7,6,2,3] => [1,7,6,5,4,3,2] => ? = 2
[2,3,7,4,5,6,1] => [6,2,3,7,4,5,1] => [1,6,5,4,7,2,3] => [1,7,6,4,5,3,2] => ? = 2
Description
The length of the longest cycle of a permutation.
Matching statistic: St000260
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 0 = 1 - 1
[1,2] => [1,2] => [2] => ([],2)
=> ? = 1 - 1
[2,1] => [2,1] => [2] => ([],2)
=> ? = 1 - 1
[1,2,3] => [1,2,3] => [3] => ([],3)
=> ? = 1 - 1
[1,3,2] => [1,3,2] => [1,2] => ([(1,2)],3)
=> ? = 1 - 1
[2,1,3] => [2,1,3] => [3] => ([],3)
=> ? = 1 - 1
[2,3,1] => [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,2] => [2,3,1] => [3] => ([],3)
=> ? = 2 - 1
[3,2,1] => [3,1,2] => [3] => ([],3)
=> ? = 1 - 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
=> ? = 1 - 1
[1,2,4,3] => [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 - 1
[1,3,2,4] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? = 1 - 1
[1,3,4,2] => [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,4,2,3] => [1,3,4,2] => [1,3] => ([(2,3)],4)
=> ? = 2 - 1
[1,4,3,2] => [1,4,2,3] => [1,3] => ([(2,3)],4)
=> ? = 1 - 1
[2,1,3,4] => [2,1,3,4] => [4] => ([],4)
=> ? = 1 - 1
[2,1,4,3] => [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 - 1
[2,3,1,4] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[2,3,4,1] => [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,1,3] => [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[2,4,3,1] => [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[3,1,2,4] => [2,3,1,4] => [4] => ([],4)
=> ? = 2 - 1
[3,1,4,2] => [3,4,1,2] => [4] => ([],4)
=> ? = 2 - 1
[3,2,1,4] => [3,1,2,4] => [4] => ([],4)
=> ? = 1 - 1
[3,2,4,1] => [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,1,2] => [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,2,1] => [4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,1,2,3] => [2,3,4,1] => [4] => ([],4)
=> ? = 2 - 1
[4,1,3,2] => [3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,2,1,3] => [3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[4,2,3,1] => [4,3,1,2] => [1,3] => ([(2,3)],4)
=> ? = 2 - 1
[4,3,1,2] => [2,4,1,3] => [4] => ([],4)
=> ? = 2 - 1
[4,3,2,1] => [4,1,2,3] => [4] => ([],4)
=> ? = 1 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
=> ? = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> ? = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,3,5,2,4] => [1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,5,4,2] => [1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[1,4,2,5,3] => [1,4,5,2,3] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[1,4,3,2,5] => [1,4,2,3,5] => [1,4] => ([(3,4)],5)
=> ? = 1 - 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,4,5,2,3] => [1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,4,5,3,2] => [1,5,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,5,2,3,4] => [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[1,5,2,4,3] => [1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,5,3,2,4] => [1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,3,4,2] => [1,5,4,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,4,2,3] => [1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[1,5,4,3,2] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ? = 1 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [5] => ([],5)
=> ? = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,1,5,3,4] => [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[2,1,5,4,3] => [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 - 1
[2,3,1,4,5] => [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,1,5,4] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,3,4,5,1] => [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,4,3,5,1] => [5,2,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,4,5,1,3] => [3,2,5,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,4,5,3,1] => [5,2,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,5,1,4,3] => [4,2,5,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,1,4,5,2] => [3,5,1,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,2,4,5,1] => [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,4,2,5,1] => [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)
=> 1 = 2 - 1
[3,4,5,1,2] => [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,4,5,2,1] => [5,1,3,4,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,5,1,4,2] => [4,5,3,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,2,3,5,1] => [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,2,5,1,3] => [3,4,5,2,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,3,1,5,2] => [4,5,1,3,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,3,2,5,1] => [5,4,2,3,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,3,5,1,2] => [2,5,4,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,3,5,2,1] => [5,1,4,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,5,1,2,3] => [2,3,5,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,5,1,3,2] => [3,5,2,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,5,2,1,3] => [3,1,5,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,5,2,3,1] => [5,3,1,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,5,3,1,2] => [2,5,1,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,5,3,2,1] => [5,1,2,4,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,1,2,4,3] => [2,4,5,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,1,3,4,2] => [3,5,4,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,1,4,2,3] => [4,3,5,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,1,4,3,2] => [4,5,2,3,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,2,1,4,3] => [4,1,5,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,2,4,3,1] => [5,4,1,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,2,3,5,6,4] => [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)
=> 1 = 2 - 1
[1,2,4,5,6,3] => [1,2,6,4,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,2,5,4,6,3] => [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)
=> 1 = 2 - 1
[1,2,5,6,3,4] => [1,2,4,6,5,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,2,5,6,4,3] => [1,2,6,3,5,4] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,2,6,3,5,4] => [1,2,5,6,4,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000455
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
[1,2] => ([],2)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,1] => ([(0,1)],2)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
[1,2,3] => ([],3)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[1,3,2] => ([(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,1,3] => ([(1,2)],3)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
[1,2,3,4] => ([],4)
=> ([],4)
=> ([],4)
=> ? = 1 - 2
[1,2,4,3] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[1,3,2,4] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,1,3,4] => ([(2,3)],4)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ([],1)
=> ? = 2 - 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 2 - 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 2 - 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 2 - 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ([],1)
=> ? = 1 - 2
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ([],5)
=> ? = 1 - 2
[1,2,3,5,4] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 1 - 2
[1,2,4,3,5] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 1 - 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 1 - 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 2 - 2
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,1,3,4,5] => ([(3,4)],5)
=> ([],4)
=> ([],4)
=> ? = 1 - 2
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ([],2)
=> ? = 1 - 2
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 0 = 2 - 2
[3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ? = 2 - 2
[3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2 - 2
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 2
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ([],3)
=> ? = 1 - 2
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 0 = 2 - 2
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0 = 2 - 2
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 2 - 2
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0 = 2 - 2
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> ? = 1 - 1
[1,2] => ([],2)
=> ([],2)
=> ([],2)
=> ? = 1 - 1
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> ? = 1 - 1
[1,2,3] => ([],3)
=> ([],3)
=> ([],3)
=> ? = 1 - 1
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ? = 1 - 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([],2)
=> ? = 1 - 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],1)
=> ? = 1 - 1
[1,2,3,4] => ([],4)
=> ([],4)
=> ([],4)
=> ? = 1 - 1
[1,2,4,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ? = 1 - 1
[1,3,2,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ? = 1 - 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 1 - 1
[2,1,3,4] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([],3)
=> ? = 1 - 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> ? = 1 - 1
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([],2)
=> ? = 1 - 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([],1)
=> ? = 2 - 1
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 - 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 - 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 2 - 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],1)
=> ? = 1 - 1
[1,2,3,4,5] => ([],5)
=> ([],5)
=> ([],5)
=> ? = 1 - 1
[1,2,3,5,4] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ? = 1 - 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ? = 1 - 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? = 1 - 1
[1,3,2,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ? = 1 - 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 - 1
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 2 - 1
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([],3)
=> ? = 1 - 1
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([],2)
=> ? = 2 - 1
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 2 - 1
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(2,3)],4)
=> ? = 2 - 1
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 2 - 1
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 2 - 1
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],2)
=> ? = 1 - 1
[2,1,3,4,5] => ([(3,4)],5)
=> ([(3,4)],5)
=> ([],4)
=> ? = 1 - 1
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 - 1
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],3)
=> ? = 1 - 1
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 2 - 1
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1 = 2 - 1
[4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 1 = 2 - 1
[4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 2 - 1
[5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 1 = 2 - 1
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: St000308
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [2,1] => [1,2] => [2,1] => 1
[2,1] => [1,2] => [1,2] => [2,1] => 1
[1,2,3] => [3,2,1] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [2,3,1] => [1,2,3] => [3,2,1] => 1
[2,1,3] => [3,1,2] => [1,2,3] => [3,2,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,3,1] => 2
[3,1,2] => [2,1,3] => [1,3,2] => [2,3,1] => 2
[3,2,1] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [4,3,2,1] => 1
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 2
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [3,4,2,1] => 2
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [4,3,2,1] => 1
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [4,2,3,1] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [3,2,4,1] => 2
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [3,2,4,1] => 2
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [2,4,3,1] => 2
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [4,2,3,1] => 2
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [4,3,2,1] => 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [3,2,4,1] => 2
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [3,2,4,1] => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 2
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [3,2,4,1] => 2
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [3,2,4,1] => 2
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [3,4,2,1] => 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [2,4,3,1] => 2
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [3,5,4,2,1] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [4,3,5,2,1] => 2
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 2
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 2
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 2
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 2
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => ? = 2
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 2
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => ? = 2
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 2
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 2
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 2
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => ? = 2
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => ? = 2
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => ? = 2
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => ? = 2
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,5,3,6,7,4] => [4,7,6,3,5,2,1] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 2
[1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => ? = 2
[1,2,5,3,7,6,4] => [4,6,7,3,5,2,1] => [1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => ? = 2
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 1
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St001235
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [2,1] => [1,2] => [2] => 1
[2,1] => [1,2] => [1,2] => [2] => 1
[1,2,3] => [3,2,1] => [1,2,3] => [3] => 1
[1,3,2] => [2,3,1] => [1,2,3] => [3] => 1
[2,1,3] => [3,1,2] => [1,2,3] => [3] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,1] => 2
[3,1,2] => [2,1,3] => [1,3,2] => [2,1] => 2
[3,2,1] => [1,2,3] => [1,2,3] => [3] => 1
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [4] => 1
[1,2,4,3] => [3,4,2,1] => [1,2,3,4] => [4] => 1
[1,3,2,4] => [4,2,3,1] => [1,2,3,4] => [4] => 1
[1,3,4,2] => [2,4,3,1] => [1,2,4,3] => [3,1] => 2
[1,4,2,3] => [3,2,4,1] => [1,2,4,3] => [3,1] => 2
[1,4,3,2] => [2,3,4,1] => [1,2,3,4] => [4] => 1
[2,1,3,4] => [4,3,1,2] => [1,2,3,4] => [4] => 1
[2,1,4,3] => [3,4,1,2] => [1,2,3,4] => [4] => 1
[2,3,1,4] => [4,1,3,2] => [1,3,2,4] => [2,2] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,2,3] => [2,2] => 2
[2,4,1,3] => [3,1,4,2] => [1,4,2,3] => [2,2] => 2
[2,4,3,1] => [1,3,4,2] => [1,3,4,2] => [3,1] => 2
[3,1,2,4] => [4,2,1,3] => [1,3,2,4] => [2,2] => 2
[3,1,4,2] => [2,4,1,3] => [1,3,2,4] => [2,2] => 2
[3,2,1,4] => [4,1,2,3] => [1,2,3,4] => [4] => 1
[3,2,4,1] => [1,4,2,3] => [1,4,2,3] => [2,2] => 2
[3,4,1,2] => [2,1,4,3] => [1,4,2,3] => [2,2] => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [3,1] => 2
[4,1,2,3] => [3,2,1,4] => [1,4,2,3] => [2,2] => 2
[4,1,3,2] => [2,3,1,4] => [1,4,2,3] => [2,2] => 2
[4,2,1,3] => [3,1,2,4] => [1,2,4,3] => [3,1] => 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,2] => 2
[4,3,1,2] => [2,1,3,4] => [1,3,4,2] => [3,1] => 2
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [4] => 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [5] => 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,2,3,4,5] => [5] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,3,4,5] => [5] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,2,3,5,4] => [4,1] => 2
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,3,5,4] => [4,1] => 2
[1,2,5,4,3] => [3,4,5,2,1] => [1,2,3,4,5] => [5] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,3,4,5] => [5] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,2,3,4,5] => [5] => 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,4,3,5] => [3,2] => 2
[1,3,4,5,2] => [2,5,4,3,1] => [1,2,5,3,4] => [3,2] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,5,3,4] => [3,2] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [1,2,4,5,3] => [4,1] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [1,2,4,3,5] => [3,2] => 2
[1,4,2,5,3] => [3,5,2,4,1] => [1,2,4,3,5] => [3,2] => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,2,3,4,5] => [5] => 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,2,5,3,4] => [3,2] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,5,3,4] => [3,2] => 2
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 2
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [1,2,3,4,6,7,5] => [6,1] => ? = 2
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 2
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 2
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [1,2,3,4,7,5,6] => [5,2] => ? = 2
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [1,2,3,4,6,5,7] => [5,2] => ? = 2
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [1,2,3,4,6,7,5] => [6,1] => ? = 2
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [1,2,3,4,5,7,6] => [6,1] => ? = 2
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 2
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 2
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [1,2,3,6,7,4,5] => [5,2] => ? = 2
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [1,2,3,5,7,4,6] => [5,2] => ? = 2
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [1,2,3,7,4,5,6] => [4,3] => ? = 2
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [1,2,3,5,7,4,6] => [5,2] => ? = 2
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [1,2,3,6,7,4,5] => [5,2] => ? = 2
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [1,2,3,5,6,7,4] => [6,1] => ? = 2
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 2
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 2
[1,2,5,3,6,4,7] => [7,4,6,3,5,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 2
[1,2,5,3,6,7,4] => [4,7,6,3,5,2,1] => [1,2,3,5,4,7,6] => [4,2,1] => ? = 2
[1,2,5,3,7,4,6] => [6,4,7,3,5,2,1] => [1,2,3,5,4,7,6] => [4,2,1] => ? = 2
[1,2,5,3,7,6,4] => [4,6,7,3,5,2,1] => [1,2,3,5,4,6,7] => [4,3] => ? = 2
[1,2,5,4,3,6,7] => [7,6,3,4,5,2,1] => [1,2,3,4,5,6,7] => [7] => ? = 1
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. 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. St000640The rank of the largest boolean interval in a poset. St001060The distinguishing index of a graph. St000699The toughness times the least common multiple of 1,. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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