Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000973
(load all 2 compositions to match this statistic)
Mapping
St000973: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => 2
[[],[]]
 => 2
[[[]]]
 => 4
[[],[],[]]
 => 2
[[],[[]]]
 => 3
[[[]],[]]
 => 3
[[[],[]]]
 => 4
[[[[]]]]
 => 6
[[],[],[],[]]
 => 2
[[],[],[[]]]
 => 3
[[],[[]],[]]
 => 2
[[],[[],[]]]
 => 3
[[],[[[]]]]
 => 4
[[[]],[],[]]
 => 3
[[[]],[[]]]
 => 4
[[[],[]],[]]
 => 3
[[[[]]],[]]
 => 4
[[[],[],[]]]
 => 4
[[[],[[]]]]
 => 5
[[[[]],[]]]
 => 5
[[[[],[]]]]
 => 6
[[[[[]]]]]
 => 8
[[],[],[],[],[]]
 => 2
[[],[],[],[[]]]
 => 3
[[],[],[[]],[]]
 => 2
[[],[],[[],[]]]
 => 3
[[],[],[[[]]]]
 => 4
[[],[[]],[],[]]
 => 2
[[],[[]],[[]]]
 => 3
[[],[[],[]],[]]
 => 2
[[],[[[]]],[]]
 => 2
[[],[[],[],[]]]
 => 3
[[],[[],[[]]]]
 => 4
[[],[[[]],[]]]
 => 3
[[],[[[],[]]]]
 => 4
[[],[[[[]]]]]
 => 5
[[[]],[],[],[]]
 => 3
[[[]],[],[[]]]
 => 4
[[[]],[[]],[]]
 => 3
[[[]],[[],[]]]
 => 4
[[[]],[[[]]]]
 => 5
[[[],[]],[],[]]
 => 3
[[[[]]],[],[]]
 => 4
[[[],[]],[[]]]
 => 4
[[[[]]],[[]]]
 => 5
[[[],[],[]],[]]
 => 3
[[[],[[]]],[]]
 => 3
[[[[]],[]],[]]
 => 4
[[[[],[]]],[]]
 => 4
[[[[[]]]],[]]
 => 5
Description
The length of the boundary of an ordered tree. This is the sum of the number of edges to the left most and the right most leaf.
Matching statistic: St000771
(load all 2 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Values
[[]]
 => [.,.]
 => [1] => ([],1)
 => 1 = 2 - 1
[[],[]]
 => [.,[.,.]]
 => [2,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[[]]]
 => [[.,.],.]
 => [1,2] => ([],2)
 => ? = 4 - 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[],[[]]]
 => [.,[[.,.],.]]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [1,3,2] => ([(1,2)],3)
 => ? ∊ {3,4,6} - 1
[[[],[]]]
 => [[.,[.,.]],.]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {3,4,6} - 1
[[[[]]]]
 => [[[.,.],.],.]
 => [1,2,3] => ([],3)
 => ? ∊ {3,4,6} - 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[],[],[[]]]
 => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[],[[],[]]]
 => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[],[[[]]]]
 => [.,[[[.,.],.],.]]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[]],[[]]]
 => [[.,.],[[.,.],.]]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[],[],[]]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[],[[]]]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[[]],[]]]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[[],[]]]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[[[]]]]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => ([],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [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 = 5 - 1
[[],[],[],[[]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [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 = 4 - 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[],[[],[]]]
 => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[],[[[]]]]
 => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[]],[[]]]
 => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[],[],[]]]
 => [.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[],[[]]]]
 => [.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[[],[[[]],[]]]
 => [.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[[],[]]]]
 => [.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
[[],[[[[]]]]]
 => [.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[]],[],[[]]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[]],[[],[]]]
 => [[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[]],[[[]]]]
 => [[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[]],[[]]]
 => [[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[]]],[[]]]
 => [[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[],[],[]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[],[[]]]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[[]],[]]]
 => [[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[[],[]]]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[[[]]]]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[]],[],[]]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[]],[[]]]]
 => [[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[],[]],[]]]
 => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[[]]],[]]]
 => [[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[],[],[]]]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[],[[]]]]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[[]],[]]]]
 => [[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[[],[]]]]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[[[]]]]]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => ([],5)
 => ? ∊ {2,2,2,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[],[],[],[],[],[]]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(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)
 => 5 = 6 - 1
[[],[],[],[],[[]]]
 => [.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => ([(0,2),(0,3),(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)
 => 4 = 5 - 1
[[],[],[],[[]],[]]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => ([(0,3),(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)
 => 3 = 4 - 1
[[],[],[],[[],[]]]
 => [.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => ([(0,3),(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)
 => 3 = 4 - 1
[[],[],[],[[[]]]]
 => [.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[[],[],[[]],[],[]]
 => [.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => ([(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 = 3 - 1
[[],[],[[]],[[]]]
 => [.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[],[[],[]],[]]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[],[[[]]],[]]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[],[[],[],[]]]
 => [.,[.,[[.,[.,[.,.]]],.]]]
 => [5,4,3,6,2,1] => ([(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 = 3 - 1
[[],[],[[],[[]]]]
 => [.,[.,[[.,[[.,.],.]],.]]]
 => [4,5,3,6,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[],[[[]],[]]]
 => [.,[.,[[[.,.],[.,.]],.]]]
 => [3,5,4,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[],[[[],[]]]]
 => [.,[.,[[[.,[.,.]],.],.]]]
 => [4,3,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[],[[[[]]]]]
 => [.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[[],[[]],[],[],[]]
 => [.,[[.,.],[.,[.,[.,.]]]]]
 => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[[],[[]],[],[[]]]
 => [.,[[.,.],[.,[[.,.],.]]]]
 => [2,5,6,4,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[[]],[[]],[]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => [2,4,6,5,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[],[[]],[[],[]]]
 => [.,[[.,.],[[.,[.,.]],.]]]
 => [2,5,4,6,3,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[],[[]],[[[]]]]
 => [.,[[.,.],[[[.,.],.],.]]]
 => [2,4,5,6,3,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[[],[]],[],[]]
 => [.,[[.,[.,.]],[.,[.,.]]]]
 => [3,2,6,5,4,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[[[]]],[],[]]
 => [.,[[[.,.],.],[.,[.,.]]]]
 => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[[],[]],[[]]]
 => [.,[[.,[.,.]],[[.,.],.]]]
 => [3,2,5,6,4,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[],[[[]]],[[]]]
 => [.,[[[.,.],.],[[.,.],.]]]
 => [2,3,5,6,4,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[[],[],[]],[]]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => [4,3,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[[],[[]]],[]]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => [3,4,2,6,5,1] => ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[],[[[]],[]],[]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[[[],[]]],[]]
 => [.,[[[.,[.,.]],.],[.,.]]]
 => [3,2,4,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[[]],[],[],[],[]]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => [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)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[[]],[],[],[[]]]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => [1,5,6,4,3,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[[]],[],[[]],[]]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => [1,4,6,5,3,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[[]],[],[[],[]]]
 => [[.,.],[.,[[.,[.,.]],.]]]
 => [1,5,4,6,3,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[[]],[],[[[]]]]
 => [[.,.],[.,[[[.,.],.],.]]]
 => [1,4,5,6,3,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[[]],[[]],[],[]]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => [1,3,6,5,4,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[[]],[[]],[[]]]
 => [[.,.],[[.,.],[[.,.],.]]]
 => [1,3,5,6,4,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[[]],[[],[]],[]]
 => [[.,.],[[.,[.,.]],[.,.]]]
 => [1,4,3,6,5,2] => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[[]],[[[]]],[]]
 => [[.,.],[[[.,.],.],[.,.]]]
 => [1,3,4,6,5,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
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: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Values
[[]]
 => [1,0]
 => [1] => ([],1)
 => 1 = 2 - 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
[[[]]]
 => [1,1,0,0]
 => [2] => ([],2)
 => ? = 4 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ? ∊ {3,4,6} - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [3] => ([],3)
 => ? ∊ {3,4,6} - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [3] => ([],3)
 => ? ∊ {3,4,6} - 1
[[],[],[],[]]
 => [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 = 4 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [4] => ([],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => ? ∊ {3,3,4,4,4,5,5,6,8} - 1
[[],[],[],[],[]]
 => [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 = 5 - 1
[[],[],[],[[]]]
 => [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)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[],[],[[]],[]]
 => [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 = 2 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[],[[]],[],[]]
 => [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 = 2 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[],[[],[]],[]]
 => [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 = 3 - 1
[[],[[[]]],[]]
 => [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 = 3 - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[]],[],[],[]]
 => [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 = 2 - 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)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[]],[[]],[]]
 => [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 = 2 - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[]],[],[]]
 => [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 = 3 - 1
[[[[]]],[],[]]
 => [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 = 3 - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[],[[]]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5] => ([],5)
 => ? ∊ {2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10} - 1
[[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(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)
 => 5 = 6 - 1
[[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[],[],[],[[[]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => ([(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)
 => 1 = 2 - 1
[[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[],[[[]]],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[],[[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[],[],[[],[[]]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[],[],[[[]],[]]]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[],[],[[[],[]]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[],[],[[[[]]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,9,9,10,12} - 1
[[],[[]],[],[],[]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,1,1,1] => ([(0,3),(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)
 => 1 = 2 - 1
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[],[[],[]],[],[]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[[[]]],[],[]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[],[[],[],[]],[]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[[],[[[]],[]],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[[],[[[],[]]],[]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[[],[[[[]]]],[]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[[[]],[],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,1,1,1] => ([(0,2),(0,3),(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)
 => 1 = 2 - 1
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[]],[[],[]],[]]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[[]],[[[]]],[]]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[[],[]],[],[],[]]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[[[]]],[],[],[]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 3 - 1
[[[],[]],[[]],[]]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[[]]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[[[],[],[]],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[[[],[[]]],[],[]]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[[[[]],[]],[],[]]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 1
[[[[],[]]],[],[]]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 4 - 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: St001875
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 33%
Values
[[]]
 => [1,0]
 => [[1],[]]
 => ([],1)
 => ? = 2
[[],[]]
 => [1,0,1,0]
 => [[1,1],[]]
 => ([],1)
 => ? ∊ {2,4}
[[[]]]
 => [1,1,0,0]
 => [[2],[]]
 => ([],1)
 => ? ∊ {2,4}
[[],[],[]]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {2,3,3,4,6}
[[],[[]]]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => ([],1)
 => ? ∊ {2,3,3,4,6}
[[[]],[]]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => ([],1)
 => ? ∊ {2,3,3,4,6}
[[[],[]]]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => ([],1)
 => ? ∊ {2,3,3,4,6}
[[[[]]]]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => ([],1)
 => ? ∊ {2,3,3,4,6}
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => ([],1)
 => ? ∊ {2,2,3,3,3,3,4,4,4,4,5,5,6,8}
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => ([],1)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[[],[[]],[[]],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
[[],[[]],[[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[[]],[[[]]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[],[[],[]],[[]]]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[[],[[[]]],[[]]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[[],[[]]],[]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[[[]],[]],[]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2,1],[2]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[[],[[]],[]]]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[[[]],[[]]]]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[[[]],[],[[]],[]]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[[[]],[[]],[],[]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[[]],[[]],[[]]]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4
[[[]],[[],[]],[]]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[[[]],[[[]]],[]]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[[]],[[],[[]]]]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [[4,4,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[[]],[[[]],[]]]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [[4,3,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[[],[]],[[]],[]]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 3
[[[[]]],[[]],[]]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[[],[]],[[],[]]]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 3
[[[[]]],[[[]]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[[],[[]]],[[]]]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [[4,3,3],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[[[]],[]],[[]]]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [[4,3,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[[[],[[]],[]],[]]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [[4,4,3],[3,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[[[]],[[]]],[]]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[[],[[]],[[]]]]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [[4,4,3],[2,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[[[]],[],[[]]]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [[4,4,2],[2]]
 => ([(0,2),(2,1)],3)
 => 3
[[[[]],[[]],[]]]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [[4,3,2],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[],[],[[]],[],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[],[],[[]],[[]]]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[],[[]],[],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2,1,1],[1,1,1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[],[[]],[],[[]]]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1,1],[1,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 4
[[],[],[[]],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1,1],[2,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5
[[],[],[[]],[[],[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3,1,1],[2,2]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[],[[],[]],[[]]]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[[],[[]],[],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [[3,2,2,2,1],[1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[[],[[]],[],[[]],[]]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [[3,3,2,2,1],[2,1,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5
[[],[[]],[],[[],[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [[4,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
[[],[[]],[[]],[],[]]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [[3,3,3,2,1],[2,2,1]]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 5
[[],[[]],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2,1],[2,1]]
 => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => 6
[[],[[]],[[],[]],[]]
 => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [[4,4,2,1],[3,1]]
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 5
[[],[[]],[[],[],[]]]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [[5,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 3
[[],[[],[]],[],[[]]]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [[4,3,3,1],[2,2]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[[],[[],[]],[[]],[]]
 => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [[4,4,3,1],[3,2]]
 => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 5
[[],[[],[]],[[],[]]]
 => [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [[5,3,1],[2]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St001880
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 42%
Values
[[]]
 => [1,0]
 => [1] => ([],1)
 => ? = 2
[[],[]]
 => [1,0,1,0]
 => [2,1] => ([],2)
 => ? ∊ {2,4}
[[[]]]
 => [1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => ? ∊ {2,4}
[[],[],[]]
 => [1,0,1,0,1,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ? ∊ {2,3,4,6}
[[],[[]]]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ? ∊ {2,3,4,6}
[[[]],[]]
 => [1,1,0,0,1,0]
 => [3,1,2] => ([(1,2)],3)
 => ? ∊ {2,3,4,6}
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,3,2] => ([(0,1),(0,2)],3)
 => ? ∊ {2,3,4,6}
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ? ∊ {2,2,3,3,3,3,4,4,5,5,6,8}
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 5
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,8,10}
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 4
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 5
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
[[[],[],[],[[]]]]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6
[[[],[[[]]],[]]]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 6
[[[],[[[]],[]]]]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 5
[[[],[[[],[]]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 4
[[[[]],[],[],[]]]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 6
[[[[]],[],[[]]]]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
 => 4
[[[[],[[]]],[]]]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
 => 5
[[[[[],[]]],[]]]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
 => 4
[[[[],[[]],[]]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 6
[[[[],[[],[]]]]]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 5
[[[[[],[]],[]]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
 => 5
[[[[[],[],[]]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 6
[[[[[[[]]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6
[[[],[],[[]],[[]]]]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 7
[[[],[],[[],[[]]]]]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
 => 6
[[[],[[],[]],[[]]]]
 => [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
 => 6
[[[],[[],[],[]],[]]]
 => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 7
[[[],[[[[]]]],[]]]
 => [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
 => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 7
[[[],[[],[],[[]]]]]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,3,4,6,2,5,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
 => 6
[[[],[[[[]]],[]]]]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => 6
[[[],[[[[]],[]]]]]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => [1,3,4,5,2,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
 => 5
[[[],[[[[],[]]]]]]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
 => 4
[[[[]],[[]],[],[]]]
 => [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 7
[[[[]],[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
 => 5
[[[[]],[[],[]],[]]]
 => [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => [1,4,2,5,6,3,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
 => 6
[[[[]],[[],[],[]]]]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => [1,4,5,2,6,3,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
 => 6
[[[[]],[[],[[]]]]]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => [1,4,5,6,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
 => 4
[[[[[]],[]],[],[]]]
 => [1,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => [1,5,2,6,3,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
 => 6
[[[[],[],[]],[[]]]]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
 => 6
[[[[[]],[]],[[]]]]
 => [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
 => 4
[[[[],[[[]]]],[]]]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [1,4,2,3,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
 => 6
[[[[[]],[],[]],[]]]
 => [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => [1,5,2,3,6,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
 => 6
[[[[[],[[]]]],[]]]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
 => 5
[[[[[[],[]]]],[]]]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
 => 4
[[[[],[],[],[[]]]]]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 7
[[[[],[[[]]],[]]]]
 => [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
 => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 7
[[[[],[[[]],[]]]]]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => 6
[[[[],[[[],[]]]]]]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
 => 5
[[[[[]],[],[],[]]]]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => [1,2,5,3,6,4,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 7
[[[[[]],[],[[]]]]]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
 => 5
[[[[[],[[]]],[]]]]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
 => 6
[[[[[[],[]]],[]]]]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
 => 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001232
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 67%
Values
[[]]
 => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[[],[]]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[[[]]]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 4 - 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[[],[[]]]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[[[]],[]]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {2,3,6} - 1
[[[],[]]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {2,3,6} - 1
[[[[]]]]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {2,3,6} - 1
[[],[],[],[]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 3 - 1
[[],[],[[]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {2,2,3,3,3,4,4,4,5,6,8} - 1
[[],[[],[]]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {2,2,3,3,3,4,4,4,5,6,8} - 1
[[],[[[]]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {2,2,3,3,3,4,4,4,5,6,8} - 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? ∊ {2,2,3,3,3,4,4,4,5,6,8} - 1
[[[]],[[]]]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? ∊ {2,2,3,3,3,4,4,4,5,6,8} - 1
[[[],[]],[]]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ? ∊ {2,2,3,3,3,4,4,4,5,6,8} - 1
[[[[]]],[]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {2,2,3,3,3,4,4,4,5,6,8} - 1
[[[],[],[]]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[[[],[[]]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,3,3,3,4,4,4,5,6,8} - 1
[[[[]],[]]]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {2,2,3,3,3,4,4,4,5,6,8} - 1
[[[[],[]]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {2,2,3,3,3,4,4,4,5,6,8} - 1
[[[[[]]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,2,3,3,3,4,4,4,5,6,8} - 1
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[[],[],[],[[]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 5 = 6 - 1
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 5 = 6 - 1
[[],[],[[],[]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[],[],[[[]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[],[[]],[],[]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 5 = 6 - 1
[[],[[]],[[]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[],[[],[]],[]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 6 = 7 - 1
[[],[[[]]],[]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[],[[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3 = 4 - 1
[[],[[],[[]]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[],[[[]],[]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[],[[[],[]]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[],[[[[]]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[]],[],[[]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[]],[[]],[]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[]],[[],[]]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[]],[[[]]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[]]],[],[]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[],[]],[[]]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[]]],[[]]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[],[[]]],[]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[]],[]],[]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[],[]]],[]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[[]]]],[]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[[],[],[[]]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 6 - 1
[[[],[[]],[]]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[],[[],[]]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[],[[[]]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[]],[],[]]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[]],[[]]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[],[]],[]]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[[]]],[]]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[],[],[]]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[],[[]]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[[]],[]]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[[],[]]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[[[[[]]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,7,8,10} - 1
[[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => 3 = 4 - 1
[[],[],[],[],[[]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => 6 = 7 - 1
[[],[],[],[[]],[]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0,1,0]
 => 5 = 6 - 1
[[],[],[],[[],[]]]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? ∊ {2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,9,10,12} - 1
[[],[],[[]],[],[]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => 7 = 8 - 1
[[],[],[[]],[[]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => 8 = 9 - 1
[[],[],[[],[]],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
 => 7 = 8 - 1
[[],[],[[],[],[]]]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => 4 = 5 - 1
[[],[[]],[],[[]]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,1,0,0]
 => 7 = 8 - 1
[[],[[],[],[],[]]]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => 5 = 6 - 1
[[],[[],[],[[]]]]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => 6 = 7 - 1
[[],[[[],[]],[]]]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => 7 = 8 - 1
[[[],[],[]],[],[]]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
 => 5 = 6 - 1
[[[],[],[],[]],[]]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
 => 6 = 7 - 1
[[[],[],[],[],[]]]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => 4 = 5 - 1
[[[],[],[],[[]]]]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => 6 = 7 - 1
[[[[],[],[],[]]]]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 6 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.