searching the database
Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001538
St001538: Decorated permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => 3
[-,+] => 1
[+,-] => -1
[-,-] => -3
[2,1] => 0
[+,+,+] => 6
[-,+,+] => 4
[+,-,+] => 2
[+,+,-] => 0
[-,-,+] => 0
[-,+,-] => -2
[+,-,-] => -4
[-,-,-] => -6
[+,3,2] => 1
[-,3,2] => -1
[2,1,+] => 3
[2,1,-] => -3
[2,3,1] => 0
[3,1,2] => 0
[3,+,1] => 2
[3,-,1] => -2
[+,+,+,+] => 10
[-,+,+,+] => 8
[+,-,+,+] => 6
[+,+,-,+] => 4
[+,+,+,-] => 2
[-,-,+,+] => 4
[-,+,-,+] => 2
[-,+,+,-] => 0
[+,-,-,+] => 0
[+,-,+,-] => -2
[+,+,-,-] => -4
[-,-,-,+] => -2
[-,-,+,-] => -4
[-,+,-,-] => -6
[+,-,-,-] => -8
[-,-,-,-] => -10
[+,+,4,3] => 3
[-,+,4,3] => 1
[+,-,4,3] => -1
[-,-,4,3] => -3
[+,3,2,+] => 5
[-,3,2,+] => 3
[+,3,2,-] => -3
[-,3,2,-] => -5
[+,3,4,2] => 1
[-,3,4,2] => -1
[+,4,2,3] => 1
[-,4,2,3] => -1
[+,4,+,2] => 4
Description
Sum of decorated fixed points as signed integers.
Every fixed point decorated with a '+' is treated as a positive integer, and every fixed point decorated with a '-' is treated as a negative integer.
Matching statistic: St000259
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[+,+] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[-,+] => [2,1] => ([(0,1)],2)
=> 1
[+,-] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[-,-] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[2,1] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[+,+,+] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[-,+,+] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[+,-,+] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[+,+,-] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[-,-,+] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[-,+,-] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[+,-,-] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[-,-,-] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[+,3,2] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[-,3,2] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[2,1,+] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[2,1,-] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[3,1,2] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[3,+,1] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[3,-,1] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,1,3,4,6}
[+,+,+,+] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,+,+] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[+,-,+,+] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,-,+] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,+,-] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,+,+] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[-,+,-,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[-,+,+,-] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,-,+] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,+,-] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,-,-] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,-,+] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[-,-,+,-] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,-,-] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,-,-] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,-,-] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,4,3] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,4,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,4,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,3,2,+] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,2,+] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 3
[+,3,2,-] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,2,-] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,3,4,2] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,4,2] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,2,3] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,2,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,+,2] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,+,2] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,-,2] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,-,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,+,+] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,-,+] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,+,-] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,-,-] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,4,3] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,4,4,5,5,6,6,7,8,10}
[4,-,+,1] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 3
[-,+,+,+,+] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[-,-,+,+,+] => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[-,+,-,+,+] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[-,+,+,-,+] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[-,-,-,+,+] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[-,-,+,-,+] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[-,+,-,-,+] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[-,-,-,-,+] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[-,+,4,3,+] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[-,-,4,3,+] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[-,3,2,+,+] => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[-,3,2,-,+] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[-,3,4,2,+] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[-,4,2,3,+] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[-,4,+,2,+] => [3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[-,4,-,2,+] => [2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[-,5,-,2,4] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[-,5,-,+,2] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[2,4,+,1,+] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[2,5,-,+,1] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[4,-,+,1,+] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[5,-,+,1,4] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[5,-,+,+,1] => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[5,+,-,+,1] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[5,-,-,+,1] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[5,-,+,-,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[5,-,4,3,1] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
[5,3,2,+,1] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 4
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 28%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 28%
Values
[+,+] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[-,+] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[+,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[-,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[2,1] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[+,+,+] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[-,+,+] => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[+,-,+] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[+,+,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[-,-,+] => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[-,+,-] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[+,-,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[-,-,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[+,3,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[-,3,2] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[2,1,+] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[2,1,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[3,+,1] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[3,-,1] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,2,4,6}
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,+,+,+] => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[+,-,+,+] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[-,+,-,+] => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[-,+,+,-] => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,-,-,+] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,-,-,+] => [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6
[-,-,+,-] => [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,+,4,3] => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,-,4,3] => [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,3,2,+] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,3,2,+] => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,4,2,3] => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,4,+,2] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,4,+,2] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[+,4,-,2] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[-,4,-,2] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[2,1,+,+] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[2,1,-,+] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[2,1,+,-] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[2,1,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[2,1,4,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,4,4,4,5,5,6,7,8,10}
[4,-,+,1] => [3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-,+,+,+,+] => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[-,-,+,+,+] => [3,4,5,1,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[-,+,-,+,+] => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[-,+,+,-,+] => [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[-,-,-,+,+] => [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[-,+,-,-,+] => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[-,-,-,-,+] => [5,1,2,3,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 10
[-,+,4,3,+] => [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[-,-,4,3,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[-,3,2,+,+] => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[-,3,2,-,+] => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[-,3,4,2,+] => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8
[-,4,2,3,+] => [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[-,4,+,2,+] => [3,2,5,1,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[-,4,-,2,+] => [2,5,1,4,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 6
[-,5,-,2,4] => [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[-,5,-,+,2] => [4,2,1,5,3] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[2,4,+,1,+] => [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[2,5,-,+,1] => [4,1,2,5,3] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[4,-,+,1,+] => [3,1,5,4,2] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[5,-,+,1,4] => [3,1,4,5,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,-,+,+,1] => [3,4,1,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[5,+,-,+,1] => [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[5,-,-,+,1] => [4,1,5,2,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[5,-,+,-,1] => [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,-,4,3,1] => [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6
[5,3,2,+,1] => [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
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: St000771
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[+,+] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[-,+] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[+,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[-,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[2,1] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[+,+,+] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,+,+] => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[+,-,+] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,+,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,-,+] => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-,+,-] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,-,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,-,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,3,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,3,2] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[2,1,+] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[2,1,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[3,+,1] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[3,-,1] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,+,+] => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[+,-,+,+] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[-,+,-,+] => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-,+,+,-] => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,-,+] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,-,+] => [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-,-,+,-] => [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,4,3] => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,4,3] => [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,3,2,+] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,2,+] => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,2,3] => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,+,2] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,+,2] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,-,2] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,-,2] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,+,+] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,-,+] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,+,-] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,4,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[4,-,+,1] => [3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,+,+,+,+] => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[-,-,+,+,+] => [3,4,5,1,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[-,+,-,+,+] => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[-,+,+,-,+] => [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[-,-,-,+,+] => [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[-,+,-,-,+] => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[-,-,-,-,+] => [5,1,2,3,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-,+,4,3,+] => [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[-,-,4,3,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[-,3,2,+,+] => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[-,3,2,-,+] => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[-,3,4,2,+] => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[-,4,2,3,+] => [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[-,4,+,2,+] => [3,2,5,1,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
[-,4,-,2,+] => [2,5,1,4,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[-,5,-,2,4] => [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[-,5,-,+,2] => [4,2,1,5,3] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
[2,4,+,1,+] => [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,5,-,+,1] => [4,1,2,5,3] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,-,+,1,+] => [3,1,5,4,2] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[5,-,+,1,4] => [3,1,4,5,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,-,+,+,1] => [3,4,1,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[5,+,-,+,1] => [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,-,-,+,1] => [4,1,5,2,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[5,-,+,-,1] => [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,-,4,3,1] => [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,3,2,+,1] => [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[+,+] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[-,+] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[+,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[-,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[2,1] => [1,2] => [1,2] => ([],2)
=> ? ∊ {-3,-1,0,3}
[+,+,+] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,+,+] => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[+,-,+] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,+,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,-,+] => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[-,+,-] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,-,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,-,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,3,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,3,2] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[2,1,+] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[2,1,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[3,1,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[3,+,1] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[3,-,1] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,+,+] => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[+,-,+,+] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,-,+] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[-,+,-,+] => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,+,+,-] => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,-,+] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,+,-] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,-,+] => [4,1,2,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[-,-,+,-] => [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,4,3] => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,4,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,4,3] => [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,3,2,+] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,2,+] => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[+,3,2,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,2,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,4,2] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,2,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,2,3] => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,+,2] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,+,2] => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,-,2] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,-,2] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,+,+] => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,-,+] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,+,-] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,4,3] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[4,-,+,1] => [3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[-,+,+,+,+] => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[-,-,+,+,+] => [3,4,5,1,2] => [5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[-,+,-,+,+] => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[-,+,+,-,+] => [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[-,-,-,+,+] => [4,5,1,2,3] => [5,3,1,4,2] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[-,+,-,-,+] => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[-,-,-,-,+] => [5,1,2,3,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[-,+,4,3,+] => [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[-,-,4,3,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[-,3,2,+,+] => [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[-,3,2,-,+] => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[-,3,4,2,+] => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[-,4,2,3,+] => [2,3,5,1,4] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[-,4,+,2,+] => [3,2,5,1,4] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
[-,4,-,2,+] => [2,5,1,4,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
[-,5,-,2,4] => [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[-,5,-,+,2] => [4,2,1,5,3] => [2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 1
[2,4,+,1,+] => [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,-,+,1] => [4,1,2,5,3] => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[4,-,+,1,+] => [3,1,5,4,2] => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[5,-,+,1,4] => [3,1,4,5,2] => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[5,-,+,+,1] => [3,4,1,5,2] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[5,+,-,+,1] => [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,-,-,+,1] => [4,1,5,2,3] => [4,2,1,5,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[5,-,+,-,1] => [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,-,4,3,1] => [3,1,5,2,4] => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,3,2,+,1] => [2,4,1,5,3] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001632
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 10%
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 10%
Values
[+,+] => [1,2] => [2,1] => ([],2)
=> ? ∊ {-3,-1,0,3}
[-,+] => [2,1] => [1,2] => ([(0,1)],2)
=> 1
[+,-] => [1,2] => [2,1] => ([],2)
=> ? ∊ {-3,-1,0,3}
[-,-] => [1,2] => [2,1] => ([],2)
=> ? ∊ {-3,-1,0,3}
[2,1] => [1,2] => [2,1] => ([],2)
=> ? ∊ {-3,-1,0,3}
[+,+,+] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,+,+] => [2,3,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[+,-,+] => [1,3,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,+,-] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,-,+] => [3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[-,+,-] => [2,1,3] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,-,-] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,-,-] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,3,2] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[-,3,2] => [2,1,3] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[2,1,+] => [1,3,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[2,1,-] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[2,3,1] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[3,1,2] => [1,2,3] => [3,2,1] => ([],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[3,+,1] => [2,1,3] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[3,-,1] => [1,3,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {-6,-4,-3,-2,-2,-1,0,0,0,0,2,3,4,6}
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,+,+] => [2,3,4,1] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 0
[+,-,+,+] => [1,3,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,-,+] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,+,+] => [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
[-,+,-,+] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[-,+,+,-] => [2,3,1,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,-,+] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,+,-] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,-,+] => [4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 0
[-,-,+,-] => [3,1,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,-,-] => [2,1,3,4] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,+,4,3] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,+,4,3] => [2,3,1,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,-,4,3] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,-,4,3] => [3,1,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,3,2,+] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,2,+] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[+,3,2,-] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,2,-] => [2,1,3,4] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,3,4,2] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,3,4,2] => [2,1,3,4] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,2,3] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,2,3] => [2,3,1,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,+,2] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,+,2] => [3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[+,4,-,2] => [1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[-,4,-,2] => [2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,+,+] => [1,3,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,-,+] => [1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,+,-] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,-,-] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[2,1,4,3] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {-10,-8,-7,-6,-6,-5,-5,-4,-4,-4,-4,-4,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,6,6,7,8,10}
[4,-,+,1] => [3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 0
[-,+,+,+,+] => [2,3,4,5,1] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[-,-,+,+,+] => [3,4,5,1,2] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[-,+,-,+,+] => [2,4,5,1,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[-,+,+,-,+] => [2,3,5,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 0
[-,-,-,+,+] => [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[-,-,+,-,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[-,+,-,-,+] => [2,5,1,3,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[-,-,-,-,+] => [5,1,2,3,4] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5)
=> 0
[-,+,4,3,+] => [2,3,5,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 0
[-,-,4,3,+] => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[-,3,2,+,+] => [2,4,5,1,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[-,3,2,-,+] => [2,5,1,3,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[-,3,4,2,+] => [2,5,1,3,4] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[-,4,2,3,+] => [2,3,5,1,4] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 0
[-,4,+,2,+] => [3,2,5,1,4] => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[-,4,-,2,+] => [2,5,1,4,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0
[-,5,-,2,4] => [2,4,1,5,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[-,5,-,+,2] => [4,2,1,5,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 0
[2,4,+,1,+] => [3,1,5,2,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0
[2,5,-,+,1] => [4,1,2,5,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[4,-,+,1,+] => [3,1,5,4,2] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[5,-,+,1,4] => [3,1,4,5,2] => [3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5)
=> 0
[5,-,+,+,1] => [3,4,1,5,2] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[5,+,-,+,1] => [2,4,1,5,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
[5,-,-,+,1] => [4,1,5,2,3] => [2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 0
[5,-,+,-,1] => [3,1,5,2,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0
[5,-,4,3,1] => [3,1,5,2,4] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 0
[5,3,2,+,1] => [2,4,1,5,3] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 0
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!