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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000217
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000217: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 0
[3,1,2,4] => 1
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 2
[3,4,2,1] => 0
[4,1,2,3] => 3
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 1
[4,3,1,2] => 2
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 1
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 0
[1,4,2,3,5] => 1
[1,4,2,5,3] => 1
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 2
Description
The number of occurrences of the pattern 312 in a permutation.
Matching statistic: St000218
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
St000218: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000218: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 0
[1,3,2] => [2,3,1] => 0
[2,1,3] => [3,1,2] => 0
[2,3,1] => [1,3,2] => 0
[3,1,2] => [2,1,3] => 1
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [3,4,2,1] => 0
[1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [2,4,3,1] => 0
[1,4,2,3] => [3,2,4,1] => 1
[1,4,3,2] => [2,3,4,1] => 0
[2,1,3,4] => [4,3,1,2] => 0
[2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [4,1,3,2] => 0
[2,3,4,1] => [1,4,3,2] => 0
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [1,3,4,2] => 0
[3,1,2,4] => [4,2,1,3] => 1
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [4,1,2,3] => 0
[3,2,4,1] => [1,4,2,3] => 0
[3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [1,2,4,3] => 0
[4,1,2,3] => [3,2,1,4] => 3
[4,1,3,2] => [2,3,1,4] => 2
[4,2,1,3] => [3,1,2,4] => 2
[4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [2,1,3,4] => 2
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [4,5,3,2,1] => 0
[1,2,4,3,5] => [5,3,4,2,1] => 0
[1,2,4,5,3] => [3,5,4,2,1] => 0
[1,2,5,3,4] => [4,3,5,2,1] => 1
[1,2,5,4,3] => [3,4,5,2,1] => 0
[1,3,2,4,5] => [5,4,2,3,1] => 0
[1,3,2,5,4] => [4,5,2,3,1] => 0
[1,3,4,2,5] => [5,2,4,3,1] => 0
[1,3,4,5,2] => [2,5,4,3,1] => 0
[1,3,5,2,4] => [4,2,5,3,1] => 1
[1,3,5,4,2] => [2,4,5,3,1] => 0
[1,4,2,3,5] => [5,3,2,4,1] => 1
[1,4,2,5,3] => [3,5,2,4,1] => 1
[1,4,3,2,5] => [5,2,3,4,1] => 0
[1,4,3,5,2] => [2,5,3,4,1] => 0
[1,4,5,2,3] => [3,2,5,4,1] => 2
Description
The number of occurrences of the pattern 213 in a permutation.
Matching statistic: St000220
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [2,1] => 0
[2,1] => [1,2] => 0
[1,2,3] => [3,2,1] => 0
[1,3,2] => [3,1,2] => 0
[2,1,3] => [2,3,1] => 0
[2,3,1] => [2,1,3] => 0
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3] => [4,3,1,2] => 0
[1,3,2,4] => [4,2,3,1] => 0
[1,3,4,2] => [4,2,1,3] => 0
[1,4,2,3] => [4,1,3,2] => 1
[1,4,3,2] => [4,1,2,3] => 0
[2,1,3,4] => [3,4,2,1] => 0
[2,1,4,3] => [3,4,1,2] => 0
[2,3,1,4] => [3,2,4,1] => 0
[2,3,4,1] => [3,2,1,4] => 0
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [3,1,2,4] => 0
[3,1,2,4] => [2,4,3,1] => 1
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [2,3,4,1] => 0
[3,2,4,1] => [2,3,1,4] => 0
[3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [2,1,3,4] => 0
[4,1,2,3] => [1,4,3,2] => 3
[4,1,3,2] => [1,4,2,3] => 2
[4,2,1,3] => [1,3,4,2] => 2
[4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [1,2,4,3] => 2
[4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [5,4,3,1,2] => 0
[1,2,4,3,5] => [5,4,2,3,1] => 0
[1,2,4,5,3] => [5,4,2,1,3] => 0
[1,2,5,3,4] => [5,4,1,3,2] => 1
[1,2,5,4,3] => [5,4,1,2,3] => 0
[1,3,2,4,5] => [5,3,4,2,1] => 0
[1,3,2,5,4] => [5,3,4,1,2] => 0
[1,3,4,2,5] => [5,3,2,4,1] => 0
[1,3,4,5,2] => [5,3,2,1,4] => 0
[1,3,5,2,4] => [5,3,1,4,2] => 1
[1,3,5,4,2] => [5,3,1,2,4] => 0
[1,4,2,3,5] => [5,2,4,3,1] => 1
[1,4,2,5,3] => [5,2,4,1,3] => 1
[1,4,3,2,5] => [5,2,3,4,1] => 0
[1,4,3,5,2] => [5,2,3,1,4] => 0
[1,4,5,2,3] => [5,2,1,4,3] => 2
Description
The number of occurrences of the pattern 132 in a permutation.
Matching statistic: St001398
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001398: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St001398: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2,1] => ([],2)
=> 0
[2,1] => [1,2] => ([(0,1)],2)
=> 0
[1,2,3] => [3,2,1] => ([],3)
=> 0
[1,3,2] => [3,1,2] => ([(1,2)],3)
=> 0
[2,1,3] => [2,3,1] => ([(1,2)],3)
=> 0
[2,3,1] => [2,1,3] => ([(0,2),(1,2)],3)
=> 0
[3,1,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,2,4,3] => [4,3,1,2] => ([(2,3)],4)
=> 0
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> 0
[1,3,4,2] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 1
[1,4,3,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 0
[2,1,3,4] => [3,4,2,1] => ([(2,3)],4)
=> 0
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 0
[2,3,1,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 1
[2,4,3,1] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[3,1,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 1
[3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 0
[3,2,4,1] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 0
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,1] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 0
[4,1,2,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> 3
[4,1,3,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,1,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,2,3,5,4] => [5,4,3,1,2] => ([(3,4)],5)
=> 0
[1,2,4,3,5] => [5,4,2,3,1] => ([(3,4)],5)
=> 0
[1,2,4,5,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 0
[1,3,2,4,5] => [5,3,4,2,1] => ([(3,4)],5)
=> 0
[1,3,2,5,4] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 0
[1,3,4,2,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5)
=> 1
[1,3,5,4,2] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,4,2,3,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> 1
[1,4,2,5,3] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> 1
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 0
[1,4,3,5,2] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,4,5,2,3] => [5,2,1,4,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[] => [] => ([],0)
=> ? = 0
Description
Number of subsets of size 3 of elements in a poset that form a "v".
For a finite poset (P,≤), this is the number of sets \{x,y,z\} \in \binom{P}{3} that form a "v"-subposet (i.e., a subposet consisting of a bottom element covered by two incomparable elements).
Matching statistic: St000219
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
St000219: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000219: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ? = 0
[1,2] => [1,2] => ? = 0
[2,1] => [2,1] => ? = 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => 0
[2,3,1] => [3,1,2] => 0
[3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,2,3] => 0
[1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,3,2] => 0
[2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,1,2,4] => 0
[2,3,4,1] => [4,1,2,3] => 0
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [4,1,3,2] => 0
[3,1,2,4] => [2,3,1,4] => 1
[3,1,4,2] => [2,4,1,3] => 1
[3,2,1,4] => [3,2,1,4] => 0
[3,2,4,1] => [4,2,1,3] => 0
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [4,3,1,2] => 0
[4,1,2,3] => [2,3,4,1] => 3
[4,1,3,2] => [2,4,3,1] => 2
[4,2,1,3] => [3,2,4,1] => 2
[4,2,3,1] => [4,2,3,1] => 1
[4,3,1,2] => [3,4,2,1] => 2
[4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,3,4] => 0
[1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,4,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,2,3,5] => 0
[1,3,4,5,2] => [1,5,2,3,4] => 0
[1,3,5,2,4] => [1,4,2,5,3] => 1
[1,3,5,4,2] => [1,5,2,4,3] => 0
[1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,3,5,2,4] => 1
[1,4,3,2,5] => [1,4,3,2,5] => 0
[1,4,3,5,2] => [1,5,3,2,4] => 0
[1,4,5,2,3] => [1,4,5,2,3] => 2
[1,4,5,3,2] => [1,5,4,2,3] => 0
[1,5,2,3,4] => [1,3,4,5,2] => 3
[1,5,2,4,3] => [1,3,5,4,2] => 2
[] => [] => ? = 0
Description
The number of occurrences of the pattern 231 in a permutation.
Matching statistic: St001898
Mp00069: Permutations —complement⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
Mp00290: Parking functions —to ordered set partition⟶ Ordered set partitions
St001898: Ordered set partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00305: Permutations —parking function⟶ Parking functions
Mp00290: Parking functions —to ordered set partition⟶ Ordered set partitions
St001898: Ordered set partitions ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [{1}] => 0
[1,2] => [2,1] => [2,1] => [{2},{1}] => 0
[2,1] => [1,2] => [1,2] => [{1},{2}] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [{3},{2},{1}] => 0
[1,3,2] => [3,1,2] => [3,1,2] => [{2},{3},{1}] => 0
[2,1,3] => [2,3,1] => [2,3,1] => [{3},{1},{2}] => 0
[2,3,1] => [2,1,3] => [2,1,3] => [{2},{1},{3}] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [{1},{3},{2}] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [{1},{2},{3}] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [{4},{3},{2},{1}] => 0
[1,2,4,3] => [4,3,1,2] => [4,3,1,2] => [{3},{4},{2},{1}] => 0
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [{4},{2},{3},{1}] => 0
[1,3,4,2] => [4,2,1,3] => [4,2,1,3] => [{3},{2},{4},{1}] => 0
[1,4,2,3] => [4,1,3,2] => [4,1,3,2] => [{2},{4},{3},{1}] => 1
[1,4,3,2] => [4,1,2,3] => [4,1,2,3] => [{2},{3},{4},{1}] => 0
[2,1,3,4] => [3,4,2,1] => [3,4,2,1] => [{4},{3},{1},{2}] => 0
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [{3},{4},{1},{2}] => 0
[2,3,1,4] => [3,2,4,1] => [3,2,4,1] => [{4},{2},{1},{3}] => 0
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [{3},{2},{1},{4}] => 0
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [{2},{4},{1},{3}] => 1
[2,4,3,1] => [3,1,2,4] => [3,1,2,4] => [{2},{3},{1},{4}] => 0
[3,1,2,4] => [2,4,3,1] => [2,4,3,1] => [{4},{1},{3},{2}] => 1
[3,1,4,2] => [2,4,1,3] => [2,4,1,3] => [{3},{1},{4},{2}] => 1
[3,2,1,4] => [2,3,4,1] => [2,3,4,1] => [{4},{1},{2},{3}] => 0
[3,2,4,1] => [2,3,1,4] => [2,3,1,4] => [{3},{1},{2},{4}] => 0
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [{2},{1},{4},{3}] => 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [{2},{1},{3},{4}] => 0
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [{1},{4},{3},{2}] => 3
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => [{1},{3},{4},{2}] => 2
[4,2,1,3] => [1,3,4,2] => [1,3,4,2] => [{1},{4},{2},{3}] => 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [{1},{3},{2},{4}] => 1
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [{1},{2},{4},{3}] => 2
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [{1},{2},{3},{4}] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [{5},{4},{3},{2},{1}] => 0
[1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,1,2] => [{4},{5},{3},{2},{1}] => 0
[1,2,4,3,5] => [5,4,2,3,1] => [5,4,2,3,1] => [{5},{3},{4},{2},{1}] => 0
[1,2,4,5,3] => [5,4,2,1,3] => [5,4,2,1,3] => [{4},{3},{5},{2},{1}] => 0
[1,2,5,3,4] => [5,4,1,3,2] => [5,4,1,3,2] => [{3},{5},{4},{2},{1}] => 1
[1,2,5,4,3] => [5,4,1,2,3] => [5,4,1,2,3] => [{3},{4},{5},{2},{1}] => 0
[1,3,2,4,5] => [5,3,4,2,1] => [5,3,4,2,1] => [{5},{4},{2},{3},{1}] => 0
[1,3,2,5,4] => [5,3,4,1,2] => [5,3,4,1,2] => [{4},{5},{2},{3},{1}] => 0
[1,3,4,2,5] => [5,3,2,4,1] => [5,3,2,4,1] => [{5},{3},{2},{4},{1}] => 0
[1,3,4,5,2] => [5,3,2,1,4] => [5,3,2,1,4] => [{4},{3},{2},{5},{1}] => 0
[1,3,5,2,4] => [5,3,1,4,2] => [5,3,1,4,2] => [{3},{5},{2},{4},{1}] => 1
[1,3,5,4,2] => [5,3,1,2,4] => [5,3,1,2,4] => [{3},{4},{2},{5},{1}] => 0
[1,4,2,3,5] => [5,2,4,3,1] => [5,2,4,3,1] => [{5},{2},{4},{3},{1}] => 1
[1,4,2,5,3] => [5,2,4,1,3] => [5,2,4,1,3] => [{4},{2},{5},{3},{1}] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => [{5},{2},{3},{4},{1}] => 0
[1,4,3,5,2] => [5,2,3,1,4] => [5,2,3,1,4] => [{4},{2},{3},{5},{1}] => 0
[1,4,5,2,3] => [5,2,1,4,3] => [5,2,1,4,3] => [{3},{2},{5},{4},{1}] => 2
[1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [{6},{5},{4},{3},{2},{1}] => ? = 0
[1,2,3,4,6,5] => [6,5,4,3,1,2] => [6,5,4,3,1,2] => [{5},{6},{4},{3},{2},{1}] => ? = 0
[1,2,3,5,4,6] => [6,5,4,2,3,1] => [6,5,4,2,3,1] => [{6},{4},{5},{3},{2},{1}] => ? = 0
[1,2,3,5,6,4] => [6,5,4,2,1,3] => [6,5,4,2,1,3] => [{5},{4},{6},{3},{2},{1}] => ? = 0
[1,2,3,6,4,5] => [6,5,4,1,3,2] => [6,5,4,1,3,2] => [{4},{6},{5},{3},{2},{1}] => ? = 1
[1,2,3,6,5,4] => [6,5,4,1,2,3] => [6,5,4,1,2,3] => [{4},{5},{6},{3},{2},{1}] => ? = 0
[1,2,4,3,5,6] => [6,5,3,4,2,1] => [6,5,3,4,2,1] => [{6},{5},{3},{4},{2},{1}] => ? = 0
[1,2,4,3,6,5] => [6,5,3,4,1,2] => [6,5,3,4,1,2] => [{5},{6},{3},{4},{2},{1}] => ? = 0
[1,2,4,5,3,6] => [6,5,3,2,4,1] => [6,5,3,2,4,1] => [{6},{4},{3},{5},{2},{1}] => ? = 0
[1,2,4,5,6,3] => [6,5,3,2,1,4] => [6,5,3,2,1,4] => [{5},{4},{3},{6},{2},{1}] => ? = 0
[1,2,4,6,3,5] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => [{4},{6},{3},{5},{2},{1}] => ? = 1
[1,2,4,6,5,3] => [6,5,3,1,2,4] => [6,5,3,1,2,4] => [{4},{5},{3},{6},{2},{1}] => ? = 0
[1,2,5,3,4,6] => [6,5,2,4,3,1] => [6,5,2,4,3,1] => [{6},{3},{5},{4},{2},{1}] => ? = 1
[1,2,5,3,6,4] => [6,5,2,4,1,3] => [6,5,2,4,1,3] => [{5},{3},{6},{4},{2},{1}] => ? = 1
[1,2,5,4,3,6] => [6,5,2,3,4,1] => [6,5,2,3,4,1] => [{6},{3},{4},{5},{2},{1}] => ? = 0
[1,2,5,4,6,3] => [6,5,2,3,1,4] => [6,5,2,3,1,4] => [{5},{3},{4},{6},{2},{1}] => ? = 0
[1,2,5,6,3,4] => [6,5,2,1,4,3] => [6,5,2,1,4,3] => [{4},{3},{6},{5},{2},{1}] => ? = 2
[1,2,5,6,4,3] => [6,5,2,1,3,4] => [6,5,2,1,3,4] => [{4},{3},{5},{6},{2},{1}] => ? = 0
[1,2,6,3,4,5] => [6,5,1,4,3,2] => [6,5,1,4,3,2] => [{3},{6},{5},{4},{2},{1}] => ? = 3
[1,2,6,3,5,4] => [6,5,1,4,2,3] => [6,5,1,4,2,3] => [{3},{5},{6},{4},{2},{1}] => ? = 2
[1,2,6,4,3,5] => [6,5,1,3,4,2] => [6,5,1,3,4,2] => [{3},{6},{4},{5},{2},{1}] => ? = 2
[1,2,6,4,5,3] => [6,5,1,3,2,4] => [6,5,1,3,2,4] => [{3},{5},{4},{6},{2},{1}] => ? = 1
[1,2,6,5,3,4] => [6,5,1,2,4,3] => [6,5,1,2,4,3] => [{3},{4},{6},{5},{2},{1}] => ? = 2
[1,2,6,5,4,3] => [6,5,1,2,3,4] => [6,5,1,2,3,4] => [{3},{4},{5},{6},{2},{1}] => ? = 0
[1,3,2,4,5,6] => [6,4,5,3,2,1] => [6,4,5,3,2,1] => [{6},{5},{4},{2},{3},{1}] => ? = 0
[1,3,2,4,6,5] => [6,4,5,3,1,2] => [6,4,5,3,1,2] => [{5},{6},{4},{2},{3},{1}] => ? = 0
[1,3,2,5,4,6] => [6,4,5,2,3,1] => [6,4,5,2,3,1] => [{6},{4},{5},{2},{3},{1}] => ? = 0
[1,3,2,5,6,4] => [6,4,5,2,1,3] => [6,4,5,2,1,3] => [{5},{4},{6},{2},{3},{1}] => ? = 0
[1,3,2,6,4,5] => [6,4,5,1,3,2] => [6,4,5,1,3,2] => [{4},{6},{5},{2},{3},{1}] => ? = 1
[1,3,2,6,5,4] => [6,4,5,1,2,3] => [6,4,5,1,2,3] => [{4},{5},{6},{2},{3},{1}] => ? = 0
[1,3,4,2,5,6] => [6,4,3,5,2,1] => [6,4,3,5,2,1] => [{6},{5},{3},{2},{4},{1}] => ? = 0
[1,3,4,2,6,5] => [6,4,3,5,1,2] => [6,4,3,5,1,2] => [{5},{6},{3},{2},{4},{1}] => ? = 0
[1,3,4,5,2,6] => [6,4,3,2,5,1] => [6,4,3,2,5,1] => [{6},{4},{3},{2},{5},{1}] => ? = 0
[1,3,4,5,6,2] => [6,4,3,2,1,5] => [6,4,3,2,1,5] => [{5},{4},{3},{2},{6},{1}] => ? = 0
[1,3,4,6,2,5] => [6,4,3,1,5,2] => [6,4,3,1,5,2] => [{4},{6},{3},{2},{5},{1}] => ? = 1
[1,3,4,6,5,2] => [6,4,3,1,2,5] => [6,4,3,1,2,5] => [{4},{5},{3},{2},{6},{1}] => ? = 0
[1,3,5,2,4,6] => [6,4,2,5,3,1] => [6,4,2,5,3,1] => [{6},{3},{5},{2},{4},{1}] => ? = 1
[1,3,5,2,6,4] => [6,4,2,5,1,3] => [6,4,2,5,1,3] => [{5},{3},{6},{2},{4},{1}] => ? = 1
[1,3,5,4,2,6] => [6,4,2,3,5,1] => [6,4,2,3,5,1] => [{6},{3},{4},{2},{5},{1}] => ? = 0
[1,3,5,4,6,2] => [6,4,2,3,1,5] => [6,4,2,3,1,5] => [{5},{3},{4},{2},{6},{1}] => ? = 0
[1,3,5,6,2,4] => [6,4,2,1,5,3] => [6,4,2,1,5,3] => [{4},{3},{6},{2},{5},{1}] => ? = 2
[1,3,5,6,4,2] => [6,4,2,1,3,5] => [6,4,2,1,3,5] => [{4},{3},{5},{2},{6},{1}] => ? = 0
[1,3,6,2,4,5] => [6,4,1,5,3,2] => [6,4,1,5,3,2] => [{3},{6},{5},{2},{4},{1}] => ? = 3
[1,3,6,2,5,4] => [6,4,1,5,2,3] => [6,4,1,5,2,3] => [{3},{5},{6},{2},{4},{1}] => ? = 2
[1,3,6,4,2,5] => [6,4,1,3,5,2] => [6,4,1,3,5,2] => [{3},{6},{4},{2},{5},{1}] => ? = 2
[1,3,6,4,5,2] => [6,4,1,3,2,5] => [6,4,1,3,2,5] => [{3},{5},{4},{2},{6},{1}] => ? = 1
[1,3,6,5,2,4] => [6,4,1,2,5,3] => [6,4,1,2,5,3] => [{3},{4},{6},{2},{5},{1}] => ? = 2
[1,3,6,5,4,2] => [6,4,1,2,3,5] => [6,4,1,2,3,5] => [{3},{4},{5},{2},{6},{1}] => ? = 0
[1,4,2,3,5,6] => [6,3,5,4,2,1] => [6,3,5,4,2,1] => [{6},{5},{2},{4},{3},{1}] => ? = 1
[1,4,2,3,6,5] => [6,3,5,4,1,2] => [6,3,5,4,1,2] => [{5},{6},{2},{4},{3},{1}] => ? = 1
Description
The number of occurrences of an 132 pattern in an ordered set partition.
An occurrence of a pattern \pi\in\mathfrak S_k ordered set partition with blocks B_1|\dots|B_\ell is a sequence of elements e_1,\dots,e_k with e_i\in B_{j_i} and j_1 < \dots < j_k order-isomorphic to \pi.
Matching statistic: St000259
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000302
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St000466
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Values
[1] => [1] => ([],1)
=> ([],1)
=> 0
[1,2] => [1,2] => ([],2)
=> ([],1)
=> 0
[2,1] => [1,2] => ([],2)
=> ([],1)
=> 0
[1,2,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[1,3,2] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,1,3] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[2,3,1] => [1,2,3] => ([],3)
=> ([],1)
=> 0
[3,1,2] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 1
[3,2,1] => [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,2,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,2,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,3,4,2] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,1,4,3] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,1,4] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,3,4,1] => [1,2,3,4] => ([],4)
=> ([],1)
=> 0
[2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 1
[4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 2
[4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,2,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,3,2,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,3,5,4,2] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,2,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,2,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,4,3,2,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,3,5,2] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,4,5,2,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,4,5,3,2] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,5,2,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[1,5,2,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,5,3,2,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[1,5,3,4,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[1,5,4,2,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[1,5,4,3,2] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,1,3,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,3,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,3,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,4,5,3] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,1,5,3,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,1,5,4,3] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,3,1,4,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,1,5,4] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,1,5] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> ([],1)
=> 0
[2,3,5,1,4] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,3,5,4,1] => [1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,1,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,1,5,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,4,3,1,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,3,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,4,5,1,3] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,4,5,3,1] => [1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[2,5,1,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 3
[2,5,1,4,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,5,3,1,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,5,3,4,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 1
[2,5,4,1,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2
[2,5,4,3,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,3,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,3,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,3,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,2,4,5,6,3] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,2,5,6,4] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,2,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,2,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[1,3,4,5,6,2] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,5,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,4,6,5] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
[2,1,3,5,4,6] => [1,2,3,4,5,6] => ([],6)
=> ([],1)
=> 0
Description
The Gutman (or modified Schultz) index of a connected graph.
This is
\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)
where d(u) is the degree of vertex u and d(u,v) is the distance between vertices u and v.
For trees on n vertices, the modified Schultz index is related to the Wiener index via S^\ast(T)=4W(T)-(n-1)(2n-1) [1].
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001875The number of simple modules with projective dimension at most 1.
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