- FindStat

Your data matches 67 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000413
(load all 2 compositions to match this statistic)

Mapping

St000413: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[]]
 => 1
[[],[]]
 => 1
[[[]]]
 => 1
[[],[],[]]
 => 1
[[],[[]]]
 => 2
[[[]],[]]
 => 2
[[[],[]]]
 => 1
[[[[]]]]
 => 1
[[],[],[],[]]
 => 1
[[],[],[[]]]
 => 3
[[],[[]],[]]
 => 3
[[],[[],[]]]
 => 2
[[],[[[]]]]
 => 2
[[[]],[],[]]
 => 3
[[[]],[[]]]
 => 1
[[[],[]],[]]
 => 2
[[[[]]],[]]
 => 2
[[[],[],[]]]
 => 1
[[[],[[]]]]
 => 2
[[[[]],[]]]
 => 2
[[[[],[]]]]
 => 1
[[[[[]]]]]
 => 1
[[],[],[],[],[]]
 => 1
[[],[],[],[[]]]
 => 4
[[],[],[[]],[]]
 => 4
[[],[],[[],[]]]
 => 3
[[],[],[[[]]]]
 => 3
[[],[[]],[],[]]
 => 4
[[],[[]],[[]]]
 => 3
[[],[[],[]],[]]
 => 3
[[],[[[]]],[]]
 => 3
[[],[[],[],[]]]
 => 2
[[],[[],[[]]]]
 => 4
[[],[[[]],[]]]
 => 4
[[],[[[],[]]]]
 => 2
[[],[[[[]]]]]
 => 2
[[[]],[],[],[]]
 => 4
[[[]],[],[[]]]
 => 3
[[[]],[[]],[]]
 => 3
[[[]],[[],[]]]
 => 2
[[[]],[[[]]]]
 => 2
[[[],[]],[],[]]
 => 3
[[[[]]],[],[]]
 => 3
[[[],[]],[[]]]
 => 2
[[[[]]],[[]]]
 => 2
[[[],[],[]],[]]
 => 2
[[[],[[]]],[]]
 => 4
[[[[]],[]],[]]
 => 4
[[[[],[]]],[]]
 => 2
[[[[[]]]],[]]
 => 2

Description

The number of ordered trees with the same underlying unordered tree.

Matching statistic: St000282

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
St000282: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[]]
 => ([(0,1)],2)
 => 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 3
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 1
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => 1
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 2
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 1
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 3
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 3
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => 2
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 4
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 4
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 2
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 4
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 3
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 2
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 3
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 3
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 2
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => 2
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 4
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => 4
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 2

Description

The size of the preimage of the map 'to poset' from Ordered trees to Posets.

Matching statistic: St000741
(load all 5 compositions to match this statistic)

Mapping

Mp00047: Ordered trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%

Values

[[]]
 => ([(0,1)],2)
 => ([],2)
 => 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ? = 1
[[[]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[[[],[]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ? = 2
[[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[],[]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,3,3}
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {1,1,3,3}
[[[],[]],[]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,3,3}
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(2,3),(2,4),(3,4)],5)
 => 2
[[[],[[]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[[[]],[]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => 1
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ? ∊ {1,1,3,3}
[[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => 1
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[],[[[],[]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[]],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[],[]],[[]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[[[],[]]],[]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
[[[],[],[[]]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[[],[[]],[]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[[],[[],[]]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[[],[[[]]]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[[]],[],[]]]
 => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[[[]],[[]]]]
 => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(2,4),(2,5),(3,4),(3,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[[],[]],[]]]
 => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 2
[[[[[]]],[]]]
 => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([(3,4),(3,5),(4,5)],6)
 => 2
[[[[],[[]]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => 1
[[[[[]],[]]]]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(3,5),(4,5)],6)
 => 1
[[[[[],[]]]]]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([(4,5)],6)
 => ? ∊ {1,1,1,2,2,2,2,3,3,4,4,4,4,4,4,4}
[[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => 1

Description

The Colin de Verdière graph invariant.

Matching statistic: St000771
(load all 5 compositions to match this statistic)

Mapping

Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%

Values

[[]]
 => [.,.]
 => [1] => ([],1)
 => 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1
[[[]]]
 => [[.,.],.]
 => [1,2] => ([],2)
 => ? = 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {1,2}
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => ? ∊ {1,2}
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,3,3}
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,3,3}
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,3,3}
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,3,3}
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => ? ∊ {1,1,1,3,3}
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 1
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[[],[],[],[]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[],[],[[]]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[],[[]],[]]]
 => [[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[],[[],[]]]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[],[[[]]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[]],[],[]]]
 => [[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[]],[[]]]]
 => [[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[],[]],[]]]
 => [[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[[]]],[]]]
 => [[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[],[],[]]]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[],[[]]]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[[]],[]]]]
 => [[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[[],[]]]]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[[[]]]]]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,4}

Description

The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums

$0$ , which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ Its eigenvalues are

$0,4,4,6$ , so the statistic is

$2$ . The path on four vertices has eigenvalues

$0, 4.7\dots, 6, 9.2\dots$ and therefore statistic

$1$ .

Matching statistic: St000939

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%

Values

[[]]
 => [1,0]
 => []
 => ?
 => ? = 1
[[],[]]
 => [1,0,1,0]
 => [1]
 => []
 => ? ∊ {1,1}
[[[]]]
 => [1,1,0,0]
 => []
 => ?
 => ? ∊ {1,1}
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1]
 => [1]
 => ? ∊ {1,1,1,2,2}
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1]
 => [1]
 => ? ∊ {1,1,1,2,2}
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2]
 => []
 => ? ∊ {1,1,1,2,2}
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1]
 => []
 => ? ∊ {1,1,1,2,2}
[[[[]]]]
 => [1,1,1,0,0,0]
 => []
 => ?
 => ? ∊ {1,1,1,2,2}
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 2
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => ? ∊ {1,2,2,2,3,3,3}
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => []
 => ? ∊ {1,2,2,2,3,3,3}
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => ? ∊ {1,2,2,2,3,3,3}
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => ? ∊ {1,2,2,2,3,3,3}
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => []
 => ? ∊ {1,2,2,2,3,3,3}
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => []
 => ? ∊ {1,2,2,2,3,3,3}
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? ∊ {1,2,2,2,3,3,3}
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => 3
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 3
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => 4
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 4
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 4
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => 4
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 4
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => 3
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 2
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 3
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 3
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 3
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [3,2]
 => 3
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 3
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 3
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 3
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3]
 => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [2,1]
 => 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => 2
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => [1]
 => ? ∊ {2,2,2,2,2,3,4,4,4}
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => []
 => ? ∊ {2,2,2,2,2,3,4,4,4}
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [2,1]
 => 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [2,1]
 => 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 2
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,1]
 => 2
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1]
 => 2
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2]
 => 1
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => [1]
 => ? ∊ {2,2,2,2,2,3,4,4,4}
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 => ? ∊ {2,2,2,2,2,3,4,4,4}
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => [1]
 => ? ∊ {2,2,2,2,2,3,4,4,4}
[[[[],[[]]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => [1]
 => ? ∊ {2,2,2,2,2,3,4,4,4}
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 => ? ∊ {2,2,2,2,2,3,4,4,4}
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1]
 => []
 => ? ∊ {2,2,2,2,2,3,4,4,4}
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? ∊ {2,2,2,2,2,3,4,4,4}

Description

The number of characters of the symmetric group whose value on the partition is positive.

Matching statistic: St001060
(load all 2 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 75%

Values

[[]]
 => [1,0]
 => [1] => ([],1)
 => ? = 1
[[],[]]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => ? ∊ {1,1}
[[[]]]
 => [1,1,0,0]
 => [1,2] => ([],2)
 => ? ∊ {1,1}
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {1,1,1}
[[],[[]]]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ? ∊ {1,1,1}
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => ? ∊ {1,1,1}
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,1,1,1}
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1}
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 2
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 3
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1}
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 2
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1}
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => ? ∊ {1,1,1,1,1}
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => 2
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 2
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 2
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 3
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => 3
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[[[],[[]]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([],5)
 => ? ∊ {1,1,1,1,1,1,3,3,3,3,3,3,4,4,4,4,4,4}

Description

The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.

Matching statistic: St000455
(load all 11 compositions to match this statistic)

Mapping

Mp00046: Ordered trees —to graph⟶ Graphs
Mp00259: Graphs —vertex addition⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 50%

Values

[[]]
 => ([(0,1)],2)
 => ([(1,2)],3)
 => 0 = 1 - 1
[[],[]]
 => ([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[[[]]]
 => ([(0,2),(1,2)],3)
 => ([(1,3),(2,3)],4)
 => 0 = 1 - 1
[[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,2,2} - 1
[[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,2,2} - 1
[[[],[]]]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,4),(2,4),(3,4)],5)
 => 0 = 1 - 1
[[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,2,2} - 1
[[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,3} - 1
[[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,3} - 1
[[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,3} - 1
[[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1 = 2 - 1
[[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,3} - 1
[[[]],[[]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1 = 2 - 1
[[[],[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,3} - 1
[[[[]]],[]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1 = 2 - 1
[[[],[],[]]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0 = 1 - 1
[[[],[[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,3} - 1
[[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,3} - 1
[[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,3} - 1
[[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1 = 2 - 1
[[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 1 - 1
[[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[],[[],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1 = 2 - 1
[[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[],[[],[]],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1 = 2 - 1
[[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[[[]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[]],[[],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[]],[[[]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[]],[],[]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1 = 2 - 1
[[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[]],[[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[]]],[[]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[],[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[[]]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[[]],[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[[],[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[[]]]],[]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 1 - 1
[[[],[],[[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[[]],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[[],[]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1 = 2 - 1
[[[],[[[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[]],[],[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[]],[[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[[],[]],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1 = 2 - 1
[[[[[]]],[]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[],[],[]]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[],[[]]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[[[]],[]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 2 - 1
[[[[[],[]]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[[[]]]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
 => ? ∊ {1,1,1,1,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St000647

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 75%

Values

[[]]
 => [1,0]
 => [1,1,0,0]
 => [2,3,1] => 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => 1
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => 2
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => 2
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => 3
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => 3
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => 2
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => 2
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => 3
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => 2
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => 2
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [7,6,1,2,3,4,5] => 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [5,6,1,2,3,7,4] => 2
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [7,4,1,2,6,3,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [5,7,1,2,6,3,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [5,4,1,2,6,7,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [7,3,1,6,2,4,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [6,3,1,5,2,7,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [7,4,1,6,2,3,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [4,3,1,7,6,2,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [6,7,1,5,2,3,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,3,1,5,6,7,2] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,6,7,1,3,4,5] => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [2,6,5,1,3,7,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [2,7,4,1,6,3,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [2,7,5,1,6,3,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [2,5,4,1,6,7,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [6,3,7,1,2,4,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [2,3,7,6,1,4,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [6,3,5,1,2,7,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [2,3,6,5,1,7,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [6,7,4,1,2,3,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [7,3,4,1,6,2,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [2,7,4,6,1,3,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [7,3,4,6,1,2,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [2,3,4,7,6,1,5] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,7,5,1,2,3,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [5,3,4,1,6,7,2] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,6,4,5,1,7,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [2,3,7,5,6,1,4] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => 2
[[[[],[[]]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [6,3,4,5,1,7,2] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,7,4,5,6,1,3] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [7,3,4,5,6,1,2] => ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4}
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 1

Description

The number of big descents of a permutation. For a permutation

$\pi$ , this is the number of indices

$i$ such that

$\pi(i)-\pi(i+1) > 1$ . The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1]. For the number of small descents, see [[St000214]].

Matching statistic: St000772
(load all 2 compositions to match this statistic)

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 100%

Values

[[]]
 => [1,0]
 => [1] => ([],1)
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 1
[[[]]]
 => [1,1,0,0]
 => [2] => ([],2)
 => ? = 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ? ∊ {1,1,2}
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [3] => ([],3)
 => ? ∊ {1,1,2}
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => ? ∊ {1,1,2}
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [4] => ([],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => ? ∊ {1,1,1,2,2,2,2,3,3}
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[],[[]]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4}

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

$\left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right).$ 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: St001520

Mapping

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 75%

Values

[[]]
 => [1,0]
 => [2,1] => [2,1] => 0 = 1 - 1
[[],[]]
 => [1,0,1,0]
 => [3,1,2] => [1,3,2] => 0 = 1 - 1
[[[]]]
 => [1,1,0,0]
 => [2,3,1] => [3,1,2] => 0 = 1 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [3,4,1,2] => 1 = 2 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [1,3,4,2] => 0 = 1 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,3,2] => 0 = 1 - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => 1 = 2 - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [2,4,5,1,3] => 2 = 3 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [3,1,4,5,2] => 0 = 1 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [2,1,5,4,3] => 0 = 1 - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,5,1,2,4] => 2 = 3 - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [1,3,2,5,4] => 0 = 1 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => 1 = 2 - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,4,2,5,3] => 1 = 2 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [1,3,4,5,2] => 1 = 2 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => 0 = 1 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [4,5,2,1,3] => 2 = 3 - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [1,4,5,3,2] => 1 = 2 - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,5,2,4,3] => 1 = 2 - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 1 = 2 - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,2,3,4,6,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [2,3,5,6,1,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [2,4,1,5,6,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [2,3,1,6,5,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [2,4,6,1,3,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,1,4,2,6,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [4,5,2,6,1,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [2,1,5,3,6,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,1,4,5,6,2] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [2,1,3,6,5,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [3,5,6,2,1,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [4,1,5,6,3,2] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [2,1,6,3,5,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,6,1,2,4,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [1,3,2,4,6,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [2,4,5,6,1,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [4,2,1,5,6,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [2,4,1,6,5,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [4,2,6,1,3,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [1,4,2,3,6,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [1,3,4,2,6,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [2,5,4,6,1,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [5,2,3,6,1,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,2,5,3,6,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [4,1,5,3,6,2] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => [1,4,5,2,6,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [1,5,2,3,6,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [1,3,4,5,6,2] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [1,2,3,6,4,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [2,5,6,3,1,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [4,1,3,6,5,2] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [4,1,6,2,5,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [4,6,2,1,3,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => [1,4,2,6,5,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => [5,3,6,2,1,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => [1,2,4,6,5,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => [1,4,5,6,3,2] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => [1,2,6,5,4,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[],[[]]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => [5,6,1,3,2,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => [1,4,6,2,5,3] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => [1,6,2,3,5,4] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [6,1,2,3,4,5] => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4} - 1

Description

The number of strict 3-descents. A '''strict 3-descent''' of a permutation

$\pi$ of

$\{1,2, \dots ,n \}$ is a pair

$(i,i+3)$ with

$i+3 \leq n$ and

$\pi(i) > \pi(i+3)$ .

The following 57 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001638The book thickness of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001964The interval resolution global dimension of a poset. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000243The number of cyclic valleys and cyclic peaks of a permutation. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St000091The descent variation of a composition. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000233The number of nestings of a set partition. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000252The number of nodes of degree 3 of a binary tree. St000365The number of double ascents of a permutation. St000650The number of 3-rises of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001470The cyclic holeyness of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001845The number of join irreducibles minus the rank of a lattice. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001488The number of corners of a skew partition. St001624The breadth of a lattice. St001651The Frankl number of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset.