Processing math: 100%

Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000771
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => ([],1)
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 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
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[[],[]],[]]
 => [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
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [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
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[]],[],[]]
 => [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)
 => 2
[[],[[],[]],[]]
 => [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
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[]],[[]],[]]
 => [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
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [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
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [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
[[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [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
[[],[],[],[[]],[]]
 => [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)
 => 3
[[],[],[[]],[],[]]
 => [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)
 => 2
[[],[],[[],[]],[]]
 => [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
[[],[],[[[]]],[]]
 => [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
[[],[[]],[],[],[]]
 => [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)
 => 3
[[],[[]],[[]],[]]
 => [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
[[],[[],[]],[],[]]
 => [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
[[],[[[]]],[],[]]
 => [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
[[],[[],[],[]],[]]
 => [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
[[],[[],[[]]],[]]
 => [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
[[],[[[]],[]],[]]
 => [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
[[],[[[],[]]],[]]
 => [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
[[],[[[[]]]],[]]
 => [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
[[[]],[],[],[],[]]
 => [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)
 => 4
[[[]],[],[[]],[]]
 => [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)
 => 2
[[[]],[[]],[],[]]
 => [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)
 => 2
[[[]],[[],[]],[]]
 => [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
[[[]],[[[]]],[]]
 => [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
[[[],[]],[],[],[]]
 => [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)
 => 3
[[[[]]],[],[],[]]
 => [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)
 => 3
[[[],[]],[[]],[]]
 => [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)
 => 2
[[[[]]],[[]],[]]
 => [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)
 => 2
[[[],[],[]],[],[]]
 => [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
[[[],[[]]],[],[]]
 => [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
[[[[]],[]],[],[]]
 => [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
[[[[],[]]],[],[]]
 => [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
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 (4−1−2−1−14−1−2−2−14−1−1−2−14). Its eigenvalues are 0,4,4,6, so the statistic is 2. The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore statistic 1.
Matching statistic: St000982
Mapping
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [.,.]
 => [1] => => ? = 1
[[],[]]
 => [[.,.],.]
 => [1,2] => 1 => 1
[[],[],[]]
 => [[[.,.],.],.]
 => [1,2,3] => 11 => 2
[[[]],[]]
 => [[.,[.,.]],.]
 => [2,1,3] => 01 => 1
[[],[],[],[]]
 => [[[[.,.],.],.],.]
 => [1,2,3,4] => 111 => 3
[[],[[]],[]]
 => [[[.,.],[.,.]],.]
 => [1,3,2,4] => 101 => 1
[[[]],[],[]]
 => [[[.,[.,.]],.],.]
 => [2,1,3,4] => 011 => 2
[[[],[]],[]]
 => [[.,[[.,.],.]],.]
 => [2,3,1,4] => 001 => 2
[[[[]]],[]]
 => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => 001 => 2
[[],[],[],[],[]]
 => [[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => 1111 => 4
[[],[],[[]],[]]
 => [[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => 1101 => 2
[[],[[]],[],[]]
 => [[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => 1011 => 2
[[],[[],[]],[]]
 => [[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => 1001 => 2
[[],[[[]]],[]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => 1001 => 2
[[[]],[],[],[]]
 => [[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => 0111 => 3
[[[]],[[]],[]]
 => [[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => 0101 => 1
[[[],[]],[],[]]
 => [[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => 0011 => 2
[[[[]]],[],[]]
 => [[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => 0011 => 2
[[[],[],[]],[]]
 => [[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => 0001 => 3
[[[],[[]]],[]]
 => [[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => 0001 => 3
[[[[]],[]],[]]
 => [[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => 0001 => 3
[[[[],[]]],[]]
 => [[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => 0001 => 3
[[[[[]]]],[]]
 => [[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => 0001 => 3
[[],[],[],[],[],[]]
 => [[[[[[.,.],.],.],.],.],.]
 => [1,2,3,4,5,6] => 11111 => 5
[[],[],[],[[]],[]]
 => [[[[[.,.],.],.],[.,.]],.]
 => [1,2,3,5,4,6] => 11101 => 3
[[],[],[[]],[],[]]
 => [[[[[.,.],.],[.,.]],.],.]
 => [1,2,4,3,5,6] => 11011 => 2
[[],[],[[],[]],[]]
 => [[[[.,.],.],[[.,.],.]],.]
 => [1,2,4,5,3,6] => 11001 => 2
[[],[],[[[]]],[]]
 => [[[[.,.],.],[.,[.,.]]],.]
 => [1,2,5,4,3,6] => 11001 => 2
[[],[[]],[],[],[]]
 => [[[[[.,.],[.,.]],.],.],.]
 => [1,3,2,4,5,6] => 10111 => 3
[[],[[]],[[]],[]]
 => [[[[.,.],[.,.]],[.,.]],.]
 => [1,3,2,5,4,6] => 10101 => 1
[[],[[],[]],[],[]]
 => [[[[.,.],[[.,.],.]],.],.]
 => [1,3,4,2,5,6] => 10011 => 2
[[],[[[]]],[],[]]
 => [[[[.,.],[.,[.,.]]],.],.]
 => [1,4,3,2,5,6] => 10011 => 2
[[],[[],[],[]],[]]
 => [[[.,.],[[[.,.],.],.]],.]
 => [1,3,4,5,2,6] => 10001 => 3
[[],[[],[[]]],[]]
 => [[[.,.],[[.,.],[.,.]]],.]
 => [1,3,5,4,2,6] => 10001 => 3
[[],[[[]],[]],[]]
 => [[[.,.],[[.,[.,.]],.]],.]
 => [1,4,3,5,2,6] => 10001 => 3
[[],[[[],[]]],[]]
 => [[[.,.],[.,[[.,.],.]]],.]
 => [1,4,5,3,2,6] => 10001 => 3
[[],[[[[]]]],[]]
 => [[[.,.],[.,[.,[.,.]]]],.]
 => [1,5,4,3,2,6] => 10001 => 3
[[[]],[],[],[],[]]
 => [[[[[.,[.,.]],.],.],.],.]
 => [2,1,3,4,5,6] => 01111 => 4
[[[]],[],[[]],[]]
 => [[[[.,[.,.]],.],[.,.]],.]
 => [2,1,3,5,4,6] => 01101 => 2
[[[]],[[]],[],[]]
 => [[[[.,[.,.]],[.,.]],.],.]
 => [2,1,4,3,5,6] => 01011 => 2
[[[]],[[],[]],[]]
 => [[[.,[.,.]],[[.,.],.]],.]
 => [2,1,4,5,3,6] => 01001 => 2
[[[]],[[[]]],[]]
 => [[[.,[.,.]],[.,[.,.]]],.]
 => [2,1,5,4,3,6] => 01001 => 2
[[[],[]],[],[],[]]
 => [[[[.,[[.,.],.]],.],.],.]
 => [2,3,1,4,5,6] => 00111 => 3
[[[[]]],[],[],[]]
 => [[[[.,[.,[.,.]]],.],.],.]
 => [3,2,1,4,5,6] => 00111 => 3
[[[],[]],[[]],[]]
 => [[[.,[[.,.],.]],[.,.]],.]
 => [2,3,1,5,4,6] => 00101 => 2
[[[[]]],[[]],[]]
 => [[[.,[.,[.,.]]],[.,.]],.]
 => [3,2,1,5,4,6] => 00101 => 2
[[[],[],[]],[],[]]
 => [[[.,[[[.,.],.],.]],.],.]
 => [2,3,4,1,5,6] => 00011 => 3
[[[],[[]]],[],[]]
 => [[[.,[[.,.],[.,.]]],.],.]
 => [2,4,3,1,5,6] => 00011 => 3
[[[[]],[]],[],[]]
 => [[[.,[[.,[.,.]],.]],.],.]
 => [3,2,4,1,5,6] => 00011 => 3
[[[[],[]]],[],[]]
 => [[[.,[.,[[.,.],.]]],.],.]
 => [3,4,2,1,5,6] => 00011 => 3
[[[[[]]]],[],[]]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => [4,3,2,1,5,6] => 00011 => 3
Description
The length of the longest constant subword.
Matching statistic: St000774
Mapping
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000774: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[[]]
 => [1,0]
 => [1] => ([],1)
 => 1
[[],[]]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => 1
[[],[],[]]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[[[]],[]]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => 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
[[],[[]],[]]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[]],[],[]]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[[],[]],[]]
 => [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
[[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [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
[[],[],[[]],[]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[],[[]],[],[]]
 => [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)
 => 2
[[],[[],[]],[]]
 => [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
[[[]],[],[],[]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[[[]],[[]],[]]
 => [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
[[[],[]],[],[]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [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
[[[],[],[]],[]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [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
[[],[],[],[],[],[]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [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
[[],[],[],[[]],[]]
 => [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)
 => 3
[[],[],[[]],[],[]]
 => [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)
 => 2
[[],[],[[],[]],[]]
 => [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
[[],[],[[[]]],[]]
 => [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
[[],[[]],[],[],[]]
 => [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)
 => 3
[[],[[]],[[]],[]]
 => [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
[[],[[],[]],[],[]]
 => [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
[[],[[[]]],[],[]]
 => [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
[[],[[],[],[]],[]]
 => [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
[[],[[],[[]]],[]]
 => [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
[[],[[[]],[]],[]]
 => [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
[[],[[[],[]]],[]]
 => [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
[[],[[[[]]]],[]]
 => [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
[[[]],[],[],[],[]]
 => [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)
 => 4
[[[]],[],[[]],[]]
 => [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)
 => 2
[[[]],[[]],[],[]]
 => [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)
 => 2
[[[]],[[],[]],[]]
 => [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
[[[]],[[[]]],[]]
 => [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
[[[],[]],[],[],[]]
 => [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)
 => 3
[[[[]]],[],[],[]]
 => [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)
 => 3
[[[],[]],[[]],[]]
 => [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)
 => 2
[[[[]]],[[]],[]]
 => [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)
 => 2
[[[],[],[]],[],[]]
 => [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
[[[],[[]]],[],[]]
 => [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
[[[[]],[]],[],[]]
 => [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
[[[[],[]]],[],[]]
 => [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
[[],[],[[],[]],[],[]]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[[],[],[[[]]],[],[]]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
Description
The maximal multiplicity of a Laplacian eigenvalue in a graph.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 83%
Values
[[]]
 => [.,.]
 => [.,.]
 => ([],1)
 => ? = 1 + 1
[[],[]]
 => [.,[.,.]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ? = 1 + 1
[[],[],[]]
 => [.,[.,[.,.]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 2 + 1
[[[]],[]]
 => [[.,.],[.,.]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 1
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 1
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 + 1
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 1
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 1
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 1
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 1
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3 + 1
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[[],[],[],[],[],[]]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 5 + 1
[[],[],[],[[]],[]]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3 + 1
[[],[],[[]],[],[]]
 => [.,[.,[[.,.],[.,[.,.]]]]]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[[],[],[[],[]],[]]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => [[.,.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[[],[],[[[]]],[]]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => [[.,.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[[],[[]],[],[],[]]
 => [.,[[.,.],[.,[.,[.,.]]]]]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 + 1
[[],[[]],[[]],[]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 1 + 1
[[],[[],[]],[],[]]
 => [.,[[.,[.,.]],[.,[.,.]]]]
 => [[.,[.,[.,.]]],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2 + 1
[[],[[[]]],[],[]]
 => [.,[[[.,.],.],[.,[.,.]]]]
 => [[.,[[.,.],.]],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2 + 1
[[],[[],[],[]],[]]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => [[.,[.,[.,[.,.]]]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 1
[[],[[],[[]]],[]]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => [[.,[.,[[.,.],.]]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 1
[[],[[[]],[]],[]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => [[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3 + 1
[[],[[[],[]]],[]]
 => [.,[[[.,[.,.]],.],[.,.]]]
 => [[.,[[.,[.,.]],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 1
[[],[[[[]]]],[]]
 => [.,[[[[.,.],.],.],[.,.]]]
 => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 1
[[[]],[],[],[],[]]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 4 + 1
[[[]],[],[[]],[]]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2 + 1
[[[]],[[]],[],[]]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2 + 1
[[[]],[[],[]],[]]
 => [[.,.],[[.,[.,.]],[.,.]]]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[[[]],[[[]]],[]]
 => [[.,.],[[[.,.],.],[.,.]]]
 => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[[[],[]],[],[],[]]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => [[[.,[.,.]],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 + 1
[[[[]]],[],[],[]]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => [[[[.,.],.],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 + 1
[[[],[]],[[]],[]]
 => [[.,[.,.]],[[.,.],[.,.]]]
 => [[[.,[.,.]],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[[[[]]],[[]],[]]
 => [[[.,.],.],[[.,.],[.,.]]]
 => [[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[[[],[],[]],[],[]]
 => [[.,[.,[.,.]]],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 1
[[[],[[]]],[],[]]
 => [[.,[[.,.],.]],[.,[.,.]]]
 => [[[.,[[.,.],.]],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 1
[[[[]],[]],[],[]]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => [[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3 + 1
[[[[],[]]],[],[]]
 => [[[.,[.,.]],.],[.,[.,.]]]
 => [[[[.,[.,.]],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 1
[[[[[]]]],[],[]]
 => [[[[.,.],.],.],[.,[.,.]]]
 => [[[[[.,.],.],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 1
[[[],[],[],[]],[]]
 => [[.,[.,[.,[.,.]]]],[.,.]]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[],[],[[]]],[]]
 => [[.,[.,[[.,.],.]]],[.,.]]
 => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[],[[]],[]],[]]
 => [[.,[[.,.],[.,.]]],[.,.]]
 => [[[.,[[.,.],[.,.]]],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 4 + 1
[[[],[[],[]]],[]]
 => [[.,[[.,[.,.]],.]],[.,.]]
 => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[],[[[]]]],[]]
 => [[.,[[[.,.],.],.]],[.,.]]
 => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[[]],[],[]],[]]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => [[[[.,.],[.,[.,.]]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 4 + 1
[[[[]],[[]]],[]]
 => [[[.,.],[[.,.],.]],[.,.]]
 => [[[[.,.],[[.,.],.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 4 + 1
[[[[],[]],[]],[]]
 => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[[.,[.,.]],[.,.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 4 + 1
[[[[[]]],[]],[]]
 => [[[[.,.],.],[.,.]],[.,.]]
 => [[[[[.,.],.],[.,.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 4 + 1
[[[[],[],[]]],[]]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[[],[[]]]],[]]
 => [[[.,[[.,.],.]],.],[.,.]]
 => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[[[]],[]]],[]]
 => [[[[.,.],[.,.]],.],[.,.]]
 => [[[[[.,.],[.,.]],.],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 4 + 1
[[[[[],[]]]],[]]
 => [[[[.,[.,.]],.],.],[.,.]]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[[[[]]]]],[]]
 => [[[[[.,.],.],.],.],[.,.]]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[[[],[],[],[],[]],[]]
 => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => [[[.,[.,[.,[.,[.,.]]]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[],[],[],[[]]],[]]
 => [[.,[.,[.,[[.,.],.]]]],[.,.]]
 => [[[.,[.,[.,[[.,.],.]]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[],[],[[],[]]],[]]
 => [[.,[.,[[.,[.,.]],.]]],[.,.]]
 => [[[.,[.,[[.,[.,.]],.]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[],[],[[[]]]],[]]
 => [[.,[.,[[[.,.],.],.]]],[.,.]]
 => [[[.,[.,[[[.,.],.],.]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[],[[],[],[]]],[]]
 => [[.,[[.,[.,[.,.]]],.]],[.,.]]
 => [[[.,[[.,[.,[.,.]]],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[],[[],[[]]]],[]]
 => [[.,[[.,[[.,.],.]],.]],[.,.]]
 => [[[.,[[.,[[.,.],.]],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[],[[[],[]]]],[]]
 => [[.,[[[.,[.,.]],.],.]],[.,.]]
 => [[[.,[[[.,[.,.]],.],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[],[[[[]]]]],[]]
 => [[.,[[[[.,.],.],.],.]],[.,.]]
 => [[[.,[[[[.,.],.],.],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[],[],[],[]]],[]]
 => [[[.,[.,[.,[.,.]]]],.],[.,.]]
 => [[[[.,[.,[.,[.,.]]]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[],[],[[]]]],[]]
 => [[[.,[.,[[.,.],.]]],.],[.,.]]
 => [[[[.,[.,[[.,.],.]]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[],[[],[]]]],[]]
 => [[[.,[[.,[.,.]],.]],.],[.,.]]
 => [[[[.,[[.,[.,.]],.]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[],[[[]]]]],[]]
 => [[[.,[[[.,.],.],.]],.],[.,.]]
 => [[[[.,[[[.,.],.],.]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[[],[],[]]]],[]]
 => [[[[.,[.,[.,.]]],.],.],[.,.]]
 => [[[[[.,[.,[.,.]]],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[[],[[]]]]],[]]
 => [[[[.,[[.,.],.]],.],.],[.,.]]
 => [[[[[.,[[.,.],.]],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[[[],[]]]]],[]]
 => [[[[[.,[.,.]],.],.],.],[.,.]]
 => [[[[[[.,[.,.]],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[[[[[[[]]]]]],[]]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
Mapping
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 83%
Values
[[]]
 => [.,.]
 => [.,.]
 => ([],1)
 => ? = 1 + 2
[[],[]]
 => [.,[.,.]]
 => [[.,.],.]
 => ([(0,1)],2)
 => ? = 1 + 2
[[],[],[]]
 => [.,[.,[.,.]]]
 => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 2 + 2
[[[]],[]]
 => [[.,.],[.,.]]
 => [[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[[],[],[],[]]
 => [.,[.,[.,[.,.]]]]
 => [[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 2
[[],[[]],[]]
 => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 2
[[[]],[],[]]
 => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 2
[[[],[]],[]]
 => [[.,[.,.]],[.,.]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[[[[]]],[]]
 => [[[.,.],.],[.,.]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[[],[],[],[],[]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 4 + 2
[[],[],[[]],[]]
 => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 2
[[],[[]],[],[]]
 => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 2
[[],[[],[]],[]]
 => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[[],[[[]]],[]]
 => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[[[]],[],[],[]]
 => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 2
[[[]],[[]],[]]
 => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[[[],[]],[],[]]
 => [[.,[.,.]],[.,[.,.]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[[[[]]],[],[]]
 => [[[.,.],.],[.,[.,.]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[[[],[],[]],[]]
 => [[.,[.,[.,.]]],[.,.]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[[[],[[]]],[]]
 => [[.,[[.,.],.]],[.,.]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[[[[]],[]],[]]
 => [[[.,.],[.,.]],[.,.]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3 + 2
[[[[],[]]],[]]
 => [[[.,[.,.]],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[[[[[]]]],[]]
 => [[[[.,.],.],.],[.,.]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[[],[],[],[],[],[]]
 => [.,[.,[.,[.,[.,[.,.]]]]]]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 5 + 2
[[],[],[],[[]],[]]
 => [.,[.,[.,[[.,.],[.,.]]]]]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3 + 2
[[],[],[[]],[],[]]
 => [.,[.,[[.,.],[.,[.,.]]]]]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[[],[],[[],[]],[]]
 => [.,[.,[[.,[.,.]],[.,.]]]]
 => [[.,.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[[],[],[[[]]],[]]
 => [.,[.,[[[.,.],.],[.,.]]]]
 => [[.,.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[[],[[]],[],[],[]]
 => [.,[[.,.],[.,[.,[.,.]]]]]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 + 2
[[],[[]],[[]],[]]
 => [.,[[.,.],[[.,.],[.,.]]]]
 => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 1 + 2
[[],[[],[]],[],[]]
 => [.,[[.,[.,.]],[.,[.,.]]]]
 => [[.,[.,[.,.]]],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2 + 2
[[],[[[]]],[],[]]
 => [.,[[[.,.],.],[.,[.,.]]]]
 => [[.,[[.,.],.]],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2 + 2
[[],[[],[],[]],[]]
 => [.,[[.,[.,[.,.]]],[.,.]]]
 => [[.,[.,[.,[.,.]]]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 2
[[],[[],[[]]],[]]
 => [.,[[.,[[.,.],.]],[.,.]]]
 => [[.,[.,[[.,.],.]]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 2
[[],[[[]],[]],[]]
 => [.,[[[.,.],[.,.]],[.,.]]]
 => [[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3 + 2
[[],[[[],[]]],[]]
 => [.,[[[.,[.,.]],.],[.,.]]]
 => [[.,[[.,[.,.]],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 2
[[],[[[[]]]],[]]
 => [.,[[[[.,.],.],.],[.,.]]]
 => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 2
[[[]],[],[],[],[]]
 => [[.,.],[.,[.,[.,[.,.]]]]]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 4 + 2
[[[]],[],[[]],[]]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2 + 2
[[[]],[[]],[],[]]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 2 + 2
[[[]],[[],[]],[]]
 => [[.,.],[[.,[.,.]],[.,.]]]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[[[]],[[[]]],[]]
 => [[.,.],[[[.,.],.],[.,.]]]
 => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[[[],[]],[],[],[]]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => [[[.,[.,.]],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 + 2
[[[[]]],[],[],[]]
 => [[[.,.],.],[.,[.,[.,.]]]]
 => [[[[.,.],.],.],[.,[.,.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 + 2
[[[],[]],[[]],[]]
 => [[.,[.,.]],[[.,.],[.,.]]]
 => [[[.,[.,.]],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[[[[]]],[[]],[]]
 => [[[.,.],.],[[.,.],[.,.]]]
 => [[[[.,.],.],[.,.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[[[],[],[]],[],[]]
 => [[.,[.,[.,.]]],[.,[.,.]]]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 2
[[[],[[]]],[],[]]
 => [[.,[[.,.],.]],[.,[.,.]]]
 => [[[.,[[.,.],.]],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 2
[[[[]],[]],[],[]]
 => [[[.,.],[.,.]],[.,[.,.]]]
 => [[[[.,.],[.,.]],.],[.,.]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3 + 2
[[[[],[]]],[],[]]
 => [[[.,[.,.]],.],[.,[.,.]]]
 => [[[[.,[.,.]],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 2
[[[[[]]]],[],[]]
 => [[[[.,.],.],.],[.,[.,.]]]
 => [[[[[.,.],.],.],.],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 2
[[[],[],[],[]],[]]
 => [[.,[.,[.,[.,.]]]],[.,.]]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[],[],[[]]],[]]
 => [[.,[.,[[.,.],.]]],[.,.]]
 => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[],[[]],[]],[]]
 => [[.,[[.,.],[.,.]]],[.,.]]
 => [[[.,[[.,.],[.,.]]],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 4 + 2
[[[],[[],[]]],[]]
 => [[.,[[.,[.,.]],.]],[.,.]]
 => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[],[[[]]]],[]]
 => [[.,[[[.,.],.],.]],[.,.]]
 => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[[]],[],[]],[]]
 => [[[.,.],[.,[.,.]]],[.,.]]
 => [[[[.,.],[.,[.,.]]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 4 + 2
[[[[]],[[]]],[]]
 => [[[.,.],[[.,.],.]],[.,.]]
 => [[[[.,.],[[.,.],.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 4 + 2
[[[[],[]],[]],[]]
 => [[[.,[.,.]],[.,.]],[.,.]]
 => [[[[.,[.,.]],[.,.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 4 + 2
[[[[[]]],[]],[]]
 => [[[[.,.],.],[.,.]],[.,.]]
 => [[[[[.,.],.],[.,.]],.],.]
 => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ? = 4 + 2
[[[[],[],[]]],[]]
 => [[[.,[.,[.,.]]],.],[.,.]]
 => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[[],[[]]]],[]]
 => [[[.,[[.,.],.]],.],[.,.]]
 => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[[[]],[]]],[]]
 => [[[[.,.],[.,.]],.],[.,.]]
 => [[[[[.,.],[.,.]],.],.],.]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? = 4 + 2
[[[[[],[]]]],[]]
 => [[[[.,[.,.]],.],.],[.,.]]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[[[[]]]]],[]]
 => [[[[[.,.],.],.],.],[.,.]]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[[[],[],[],[],[]],[]]
 => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => [[[.,[.,[.,[.,[.,.]]]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[],[],[],[[]]],[]]
 => [[.,[.,[.,[[.,.],.]]]],[.,.]]
 => [[[.,[.,[.,[[.,.],.]]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[],[],[[],[]]],[]]
 => [[.,[.,[[.,[.,.]],.]]],[.,.]]
 => [[[.,[.,[[.,[.,.]],.]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[],[],[[[]]]],[]]
 => [[.,[.,[[[.,.],.],.]]],[.,.]]
 => [[[.,[.,[[[.,.],.],.]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[],[[],[],[]]],[]]
 => [[.,[[.,[.,[.,.]]],.]],[.,.]]
 => [[[.,[[.,[.,[.,.]]],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[],[[],[[]]]],[]]
 => [[.,[[.,[[.,.],.]],.]],[.,.]]
 => [[[.,[[.,[[.,.],.]],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[],[[[],[]]]],[]]
 => [[.,[[[.,[.,.]],.],.]],[.,.]]
 => [[[.,[[[.,[.,.]],.],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[],[[[[]]]]],[]]
 => [[.,[[[[.,.],.],.],.]],[.,.]]
 => [[[.,[[[[.,.],.],.],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[],[],[],[]]],[]]
 => [[[.,[.,[.,[.,.]]]],.],[.,.]]
 => [[[[.,[.,[.,[.,.]]]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[],[],[[]]]],[]]
 => [[[.,[.,[[.,.],.]]],.],[.,.]]
 => [[[[.,[.,[[.,.],.]]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[],[[],[]]]],[]]
 => [[[.,[[.,[.,.]],.]],.],[.,.]]
 => [[[[.,[[.,[.,.]],.]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[],[[[]]]]],[]]
 => [[[.,[[[.,.],.],.]],.],[.,.]]
 => [[[[.,[[[.,.],.],.]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[[],[],[]]]],[]]
 => [[[[.,[.,[.,.]]],.],.],[.,.]]
 => [[[[[.,[.,[.,.]]],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[[],[[]]]]],[]]
 => [[[[.,[[.,.],.]],.],.],[.,.]]
 => [[[[[.,[[.,.],.]],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[[[],[]]]]],[]]
 => [[[[[.,[.,.]],.],.],.],[.,.]]
 => [[[[[[.,[.,.]],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[[[[[[[]]]]]],[]]
 => [[[[[[.,.],.],.],.],.],[.,.]]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.