searching the database
Your data matches 8 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: St000205
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 0
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 0
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 0
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
=> [4,2,2]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,3,2]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,1,0,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,1,1,1,0,0,0,1,1,0,0,0]
=> [3,3]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> 0
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> 3
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2]
=> 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [4,4]
=> 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2]
=> 2
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000260
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%
Values
[1,1,0,0]
=> [1,2] => ([],2)
=> ? = 0 + 2
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 0 + 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 2 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 3 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 4 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 5 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 4 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 3 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,1,6,7,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [3,4,6,1,2,7,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,6,1,7,2,5] => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [3,4,7,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,5,6,7,2,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [3,5,1,6,7,2,4] => ([(0,1),(0,6),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,5,6,1,2,7,4] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,7,1,2,4,6] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,1,5,7,2,3,6] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [4,5,1,7,2,3,6] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,5,7,1,2,3,6] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [4,1,6,7,2,3,5] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,6,1,7,2,3,5] => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,6,7,1,2,3,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,1,2,6,7,3,4] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,7,3,4] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,1,6,7,2,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 2 = 0 + 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000772
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%
Values
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 0 + 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? = 0 + 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 0 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 5 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [5,3,4,1,2,6] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [3,5,1,2,4,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 3 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,6,4] => [2,1,6,4,5,3] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,6,3,4] => [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,4,6,2,5] => [1,5,6,2,3,4] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => [6,2,5,1,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => [3,6,2,4,1,5] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => [3,6,2,5,1,4] => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [4,7,1,2,3,5,6] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => [7,3,6,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => [3,6,7,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,6,7,1,3,5] => [3,5,7,1,2,4,6] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,1,6,7,3,4] => [4,7,3,5,1,2,6] => ([(0,6),(1,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => [4,7,3,6,1,2,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => [3,4,7,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => [7,2,5,1,3,4,6] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,6] => [6,7,2,5,1,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [3,4,6,1,2,7,5] => [2,7,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,6,1,7,2,5] => [5,7,2,6,1,3,4] => ([(0,3),(0,4),(0,6),(1,3),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [3,4,7,1,2,5,6] => [2,5,6,7,1,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,5,6,7,2,4] => [4,7,2,3,1,5,6] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [3,5,1,6,7,2,4] => [4,7,2,5,1,3,6] => ([(0,6),(1,2),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,5,6,1,2,7,4] => [2,7,4,6,1,3,5] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,7,1,2,4,6] => [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => [2,4,5,7,1,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => [7,3,6,2,4,1,5] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,1,5,7,2,3,6] => [3,6,7,2,4,1,5] => ([(0,4),(0,5),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => [7,3,6,2,5,1,4] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => [2,7,3,6,1,4,5] => ([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [4,5,1,7,2,3,6] => [3,6,7,2,5,1,4] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,5,7,1,2,3,6] => [2,3,6,7,1,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [4,1,6,7,2,3,5] => [3,5,7,2,4,1,6] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,6,1,7,2,3,5] => [3,5,7,2,6,1,4] => ([(0,5),(0,6),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,6,7,1,2,3,5] => [2,3,5,7,1,4,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,1,2,6,7,3,4] => [2,4,7,3,5,1,6] => ([(0,6),(1,5),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,7,3,4] => [4,7,3,6,2,5,1] => ([(0,3),(0,5),(0,6),(1,2),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,1,6,7,2,3,4] => [3,4,7,2,5,1,6] => ([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => [2,4,7,3,6,1,5] => ([(0,6),(1,3),(1,6),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => [3,4,7,2,6,1,5] => ([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1 = 0 + 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 0 + 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: St000259
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 17%
Values
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 0 + 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? = 0 + 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 0 + 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? = 2 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 3 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? = 4 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? = 5 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? = 4 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,4,3,2] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ? = 3 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? = 2 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [5,4,3,2,1,6] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => [5,3,2,1,4,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => [2,1,6,4,3,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => [5,4,2,1,3,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => [6,2,1,5,4,3] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => [5,2,1,4,3,6] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => [4,2,1,3,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,6,4] => [2,1,6,5,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,6,3,4] => [2,1,6,3,5,4] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => [5,2,1,3,4,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,4,6,2,5] => [1,6,4,3,2,5] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => [6,5,1,4,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => [6,1,4,3,2,5] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => [6,1,5,3,2,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => [6,1,2,5,4,3] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => [6,1,5,4,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => [6,1,5,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => [7,3,2,1,6,5,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => [7,6,2,1,5,4,3] => ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,7,1,3,6] => [7,2,1,5,4,3,6] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,6,7,1,3,5] => [7,2,1,6,4,3,5] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,1,6,7,3,4] => [7,2,1,3,6,5,4] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => [7,2,1,6,5,3,4] => ([(0,1),(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => [7,2,1,6,3,5,4] => ([(0,1),(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => [7,6,5,1,4,3,2] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,6] => [7,5,1,4,3,2,6] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [3,4,6,1,2,7,5] => [7,6,1,4,3,2,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [3,4,6,1,7,2,5] => [7,1,4,3,2,6,5] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [3,4,7,1,2,5,6] => [7,1,4,3,2,5,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,5,6,7,2,4] => [7,3,1,2,6,5,4] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [3,5,1,6,7,2,4] => [7,1,3,2,6,5,4] => ([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,5,6,1,2,7,4] => [7,6,1,5,3,2,4] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [3,5,7,1,2,4,6] => [7,1,5,3,2,4,6] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => [7,1,6,3,2,4,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,5,6,2,7,3] => [7,6,1,2,5,4,3] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,1,5,7,2,3,6] => [7,1,2,5,4,3,6] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3] => [7,6,1,5,4,2,3] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => [7,6,1,5,2,4,3] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => [7,1,6,2,5,4,3] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [4,5,1,7,2,3,6] => [7,1,5,4,2,3,6] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,5,7,1,2,3,6] => [7,1,5,2,4,3,6] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [4,1,6,7,2,3,5] => [7,1,2,6,4,3,5] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [4,6,1,7,2,3,5] => [7,1,6,4,2,3,5] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,6,7,1,2,3,5] => [7,1,6,2,4,3,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,1,2,6,7,3,4] => [7,1,2,3,6,5,4] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [5,1,6,2,7,3,4] => [7,1,2,6,5,3,4] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,1,6,7,2,3,4] => [7,1,2,6,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => [7,1,6,5,2,3,4] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => [7,1,6,2,3,5,4] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => [7,1,6,2,5,3,4] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2 = 0 + 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001545
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 17%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 17%
Values
[1,1,0,0]
=> [1,2] => ([],2)
=> ([],1)
=> ? = 0 + 2
[1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ([],1)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ([],1)
=> ? = 2 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 4 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 5 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ? = 4 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,3),(1,2)],4)
=> ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 3 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 2 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 3 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [2,4,5,1,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
=> ([(0,1),(2,5),(3,4),(4,5)],6)
=> ? = 2 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(1,2)],4)
=> ? = 2 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,4,6,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [2,4,5,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [2,5,1,6,7,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [2,5,6,1,7,3,4] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [2,5,6,7,1,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,5,1,6,7,2] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,5,1,7,2,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [3,4,6,1,2,7,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [3,4,7,1,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,5,6,7,2,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [3,5,6,1,2,7,4] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [3,6,7,1,2,4,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,7,1,2,3] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [4,5,1,7,2,3,6] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,5,7,1,2,3,6] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [4,1,6,7,2,3,5] => ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [4,6,7,1,2,3,5] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,1,2,6,7,3,4] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [5,1,6,7,2,3,4] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [5,6,1,2,7,3,4] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [5,6,1,7,2,3,4] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ([(0,1)],2)
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7)],8)
=> ([(0,1)],2)
=> 2 = 0 + 2
Description
The second Elser number of a connected graph.
For a connected graph $G$ the $k$-th Elser number is
$$
els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k
$$
where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$.
It is clear that this number is even. It was shown in [1] that it is non-negative.
Matching statistic: St001232
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 17%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 17%
Values
[1,1,0,0]
=> [2,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 0 + 2
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 0 + 2
[1,1,1,0,0,0]
=> [3,1,2] => [2,1]
=> [1,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 1 + 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> ? = 2 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 4 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 5 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 4 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 0 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 3 + 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 2
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 + 2
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 + 2
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 2
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2 + 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 2
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,2,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 2
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,2,4,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 3 + 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,6,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,2,3,5] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 2
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,6,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,1,4,5,6,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,1,5,6,3,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,1,3,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,1,5,6,2,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [3,4,1,2,6,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,1,6,2,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,1,5,2,6,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1,6,7,8,3,4,5] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [2,6,7,8,1,3,4,5] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,7,4,8,6] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [3,5,6,8,1,2,4,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,0,1,1,0,1,0,1,0,0,0,0]
=> [4,1,6,7,8,2,3,5] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
=> [4,5,6,1,8,2,3,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [4,5,6,8,1,2,3,7] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,0,1,1,0,0,1,0,1,0,0,0,0]
=> [4,6,1,7,8,2,3,5] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [5,1,6,7,2,8,3,4] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [5,1,6,7,8,2,3,4] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [5,6,1,7,8,2,3,4] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> [5,6,7,1,2,3,8,4] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [5,6,7,8,1,2,3,4] => [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 0 + 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000689
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[1,1,0,0]
=> [2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [5,9,1,2,3,8,4,6,7] => [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [8,4,1,2,9,3,5,6,7] => [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Matching statistic: St001001
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 17%
Values
[1,1,0,0]
=> [2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,0]
=> [3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 4
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 5
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 3
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,1,2,3,7,4,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [6,1,2,5,3,7,4] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [3,1,7,2,4,5,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [3,1,7,2,6,4,5] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,1,4,2,3,7,5] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,1,4,2,7,3,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [3,1,7,5,2,4,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,7,1,3,4,5,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,7,1,3,6,4,5] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,7,1,5,3,4,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,7,1,6,3,4,5] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 3
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 0
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,1,2,3,7,4] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,7,1,2,6,3,4] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [6,7,1,5,2,3,4] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,6,7,1,3,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [2,7,5,1,6,3,4] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,3,7,1,2,4,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,7,4,1,2,3,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,7,4,6,1,3,5] => [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [5,9,1,2,3,8,4,6,7] => [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [8,4,1,2,9,3,5,6,7] => [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
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!