Your data matches 18 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000904
(load all 12 compositions to match this statistic)
Mapping
St000904: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 3
[1,2] => 1
[2,1] => 1
[3] => 1
[1,1,1,1] => 4
[1,1,2] => 2
[1,2,1] => 2
[1,3] => 1
[2,1,1] => 2
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 5
[1,1,1,2] => 3
[1,1,2,1] => 3
[1,1,3] => 2
[1,2,1,1] => 3
[1,2,2] => 2
[1,3,1] => 2
[1,4] => 1
[2,1,1,1] => 3
[2,1,2] => 2
[2,2,1] => 2
[2,3] => 1
[3,1,1] => 2
[3,2] => 1
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 6
[1,1,1,1,2] => 4
[1,1,1,2,1] => 4
[1,1,1,3] => 3
[1,1,2,1,1] => 4
[1,1,2,2] => 2
[1,1,3,1] => 3
[1,1,4] => 2
[1,2,1,1,1] => 4
[1,2,1,2] => 2
[1,2,2,1] => 2
[1,2,3] => 1
[1,3,1,1] => 3
[1,3,2] => 1
[1,4,1] => 2
[1,5] => 1
[2,1,1,1,1] => 4
[2,1,1,2] => 2
[2,1,2,1] => 2
Description
The maximal number of repetitions of an integer composition.
Matching statistic: St001933
(load all 2 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 1
[1,1] => [1,1]
 => 2
[2] => [2]
 => 1
[1,1,1] => [1,1,1]
 => 3
[1,2] => [2,1]
 => 1
[2,1] => [2,1]
 => 1
[3] => [3]
 => 1
[1,1,1,1] => [1,1,1,1]
 => 4
[1,1,2] => [2,1,1]
 => 2
[1,2,1] => [2,1,1]
 => 2
[1,3] => [3,1]
 => 1
[2,1,1] => [2,1,1]
 => 2
[2,2] => [2,2]
 => 2
[3,1] => [3,1]
 => 1
[4] => [4]
 => 1
[1,1,1,1,1] => [1,1,1,1,1]
 => 5
[1,1,1,2] => [2,1,1,1]
 => 3
[1,1,2,1] => [2,1,1,1]
 => 3
[1,1,3] => [3,1,1]
 => 2
[1,2,1,1] => [2,1,1,1]
 => 3
[1,2,2] => [2,2,1]
 => 2
[1,3,1] => [3,1,1]
 => 2
[1,4] => [4,1]
 => 1
[2,1,1,1] => [2,1,1,1]
 => 3
[2,1,2] => [2,2,1]
 => 2
[2,2,1] => [2,2,1]
 => 2
[2,3] => [3,2]
 => 1
[3,1,1] => [3,1,1]
 => 2
[3,2] => [3,2]
 => 1
[4,1] => [4,1]
 => 1
[5] => [5]
 => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 6
[1,1,1,1,2] => [2,1,1,1,1]
 => 4
[1,1,1,2,1] => [2,1,1,1,1]
 => 4
[1,1,1,3] => [3,1,1,1]
 => 3
[1,1,2,1,1] => [2,1,1,1,1]
 => 4
[1,1,2,2] => [2,2,1,1]
 => 2
[1,1,3,1] => [3,1,1,1]
 => 3
[1,1,4] => [4,1,1]
 => 2
[1,2,1,1,1] => [2,1,1,1,1]
 => 4
[1,2,1,2] => [2,2,1,1]
 => 2
[1,2,2,1] => [2,2,1,1]
 => 2
[1,2,3] => [3,2,1]
 => 1
[1,3,1,1] => [3,1,1,1]
 => 3
[1,3,2] => [3,2,1]
 => 1
[1,4,1] => [4,1,1]
 => 2
[1,5] => [5,1]
 => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => 4
[2,1,1,2] => [2,2,1,1]
 => 2
[2,1,2,1] => [2,2,1,1]
 => 2
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000392
(load all 3 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 10 => 1
[1,1] => [1,1]
 => 110 => 2
[2] => [2]
 => 100 => 1
[1,1,1] => [1,1,1]
 => 1110 => 3
[1,2] => [2,1]
 => 1010 => 1
[2,1] => [2,1]
 => 1010 => 1
[3] => [3]
 => 1000 => 1
[1,1,1,1] => [1,1,1,1]
 => 11110 => 4
[1,1,2] => [2,1,1]
 => 10110 => 2
[1,2,1] => [2,1,1]
 => 10110 => 2
[1,3] => [3,1]
 => 10010 => 1
[2,1,1] => [2,1,1]
 => 10110 => 2
[2,2] => [2,2]
 => 1100 => 2
[3,1] => [3,1]
 => 10010 => 1
[4] => [4]
 => 10000 => 1
[1,1,1,1,1] => [1,1,1,1,1]
 => 111110 => 5
[1,1,1,2] => [2,1,1,1]
 => 101110 => 3
[1,1,2,1] => [2,1,1,1]
 => 101110 => 3
[1,1,3] => [3,1,1]
 => 100110 => 2
[1,2,1,1] => [2,1,1,1]
 => 101110 => 3
[1,2,2] => [2,2,1]
 => 11010 => 2
[1,3,1] => [3,1,1]
 => 100110 => 2
[1,4] => [4,1]
 => 100010 => 1
[2,1,1,1] => [2,1,1,1]
 => 101110 => 3
[2,1,2] => [2,2,1]
 => 11010 => 2
[2,2,1] => [2,2,1]
 => 11010 => 2
[2,3] => [3,2]
 => 10100 => 1
[3,1,1] => [3,1,1]
 => 100110 => 2
[3,2] => [3,2]
 => 10100 => 1
[4,1] => [4,1]
 => 100010 => 1
[5] => [5]
 => 100000 => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 1111110 => 6
[1,1,1,1,2] => [2,1,1,1,1]
 => 1011110 => 4
[1,1,1,2,1] => [2,1,1,1,1]
 => 1011110 => 4
[1,1,1,3] => [3,1,1,1]
 => 1001110 => 3
[1,1,2,1,1] => [2,1,1,1,1]
 => 1011110 => 4
[1,1,2,2] => [2,2,1,1]
 => 110110 => 2
[1,1,3,1] => [3,1,1,1]
 => 1001110 => 3
[1,1,4] => [4,1,1]
 => 1000110 => 2
[1,2,1,1,1] => [2,1,1,1,1]
 => 1011110 => 4
[1,2,1,2] => [2,2,1,1]
 => 110110 => 2
[1,2,2,1] => [2,2,1,1]
 => 110110 => 2
[1,2,3] => [3,2,1]
 => 101010 => 1
[1,3,1,1] => [3,1,1,1]
 => 1001110 => 3
[1,3,2] => [3,2,1]
 => 101010 => 1
[1,4,1] => [4,1,1]
 => 1000110 => 2
[1,5] => [5,1]
 => 1000010 => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => 1011110 => 4
[2,1,1,2] => [2,2,1,1]
 => 110110 => 2
[2,1,2,1] => [2,2,1,1]
 => 110110 => 2
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
(load all 3 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => 10 => 1
[1,1] => [1,1]
 => 110 => 2
[2] => [2]
 => 100 => 1
[1,1,1] => [1,1,1]
 => 1110 => 3
[1,2] => [2,1]
 => 1010 => 1
[2,1] => [2,1]
 => 1010 => 1
[3] => [3]
 => 1000 => 1
[1,1,1,1] => [1,1,1,1]
 => 11110 => 4
[1,1,2] => [2,1,1]
 => 10110 => 2
[1,2,1] => [2,1,1]
 => 10110 => 2
[1,3] => [3,1]
 => 10010 => 1
[2,1,1] => [2,1,1]
 => 10110 => 2
[2,2] => [2,2]
 => 1100 => 2
[3,1] => [3,1]
 => 10010 => 1
[4] => [4]
 => 10000 => 1
[1,1,1,1,1] => [1,1,1,1,1]
 => 111110 => 5
[1,1,1,2] => [2,1,1,1]
 => 101110 => 3
[1,1,2,1] => [2,1,1,1]
 => 101110 => 3
[1,1,3] => [3,1,1]
 => 100110 => 2
[1,2,1,1] => [2,1,1,1]
 => 101110 => 3
[1,2,2] => [2,2,1]
 => 11010 => 2
[1,3,1] => [3,1,1]
 => 100110 => 2
[1,4] => [4,1]
 => 100010 => 1
[2,1,1,1] => [2,1,1,1]
 => 101110 => 3
[2,1,2] => [2,2,1]
 => 11010 => 2
[2,2,1] => [2,2,1]
 => 11010 => 2
[2,3] => [3,2]
 => 10100 => 1
[3,1,1] => [3,1,1]
 => 100110 => 2
[3,2] => [3,2]
 => 10100 => 1
[4,1] => [4,1]
 => 100010 => 1
[5] => [5]
 => 100000 => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 1111110 => 6
[1,1,1,1,2] => [2,1,1,1,1]
 => 1011110 => 4
[1,1,1,2,1] => [2,1,1,1,1]
 => 1011110 => 4
[1,1,1,3] => [3,1,1,1]
 => 1001110 => 3
[1,1,2,1,1] => [2,1,1,1,1]
 => 1011110 => 4
[1,1,2,2] => [2,2,1,1]
 => 110110 => 2
[1,1,3,1] => [3,1,1,1]
 => 1001110 => 3
[1,1,4] => [4,1,1]
 => 1000110 => 2
[1,2,1,1,1] => [2,1,1,1,1]
 => 1011110 => 4
[1,2,1,2] => [2,2,1,1]
 => 110110 => 2
[1,2,2,1] => [2,2,1,1]
 => 110110 => 2
[1,2,3] => [3,2,1]
 => 101010 => 1
[1,3,1,1] => [3,1,1,1]
 => 1001110 => 3
[1,3,2] => [3,2,1]
 => 101010 => 1
[1,4,1] => [4,1,1]
 => 1000110 => 2
[1,5] => [5,1]
 => 1000010 => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => 1011110 => 4
[2,1,1,2] => [2,2,1,1]
 => 110110 => 2
[2,1,2,1] => [2,2,1,1]
 => 110110 => 2
Description
The length of a longest cyclic run of ones of a binary word. Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St000757
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000757: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [1] => 1
[1,1] => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[2] => [2]
 => [[1,2]]
 => [2] => 1
[1,1,1] => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 3
[1,2] => [2,1]
 => [[1,2],[3]]
 => [2,1] => 1
[2,1] => [2,1]
 => [[1,2],[3]]
 => [2,1] => 1
[3] => [3]
 => [[1,2,3]]
 => [3] => 1
[1,1,1,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 4
[1,1,2] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[1,2,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[1,3] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1
[2,1,1] => [2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[2,2] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[3,1] => [3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1
[4] => [4]
 => [[1,2,3,4]]
 => [4] => 1
[1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 5
[1,1,1,2] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 3
[1,1,2,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 3
[1,1,3] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 2
[1,2,1,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 3
[1,2,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2
[1,3,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 2
[1,4] => [4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => 1
[2,1,1,1] => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 3
[2,1,2] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2
[2,2,1] => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2
[2,3] => [3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => 1
[3,1,1] => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 2
[3,2] => [3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => 1
[4,1] => [4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => 1
[5] => [5]
 => [[1,2,3,4,5]]
 => [5] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 6
[1,1,1,1,2] => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => 4
[1,1,1,2,1] => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => 4
[1,1,1,3] => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [3,1,1,1] => 3
[1,1,2,1,1] => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => 4
[1,1,2,2] => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => 2
[1,1,3,1] => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [3,1,1,1] => 3
[1,1,4] => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [4,1,1] => 2
[1,2,1,1,1] => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => 4
[1,2,1,2] => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => 2
[1,2,2,1] => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => 2
[1,2,3] => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [3,2,1] => 1
[1,3,1,1] => [3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [3,1,1,1] => 3
[1,3,2] => [3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [3,2,1] => 1
[1,4,1] => [4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [4,1,1] => 2
[1,5] => [5,1]
 => [[1,2,3,4,5],[6]]
 => [5,1] => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => 4
[2,1,1,2] => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => 2
[2,1,2,1] => [2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => 2
Description
The length of the longest weakly inreasing subsequence of parts of an integer composition.
Matching statistic: St000899
(load all 2 compositions to match this statistic)
Mapping
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000899: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
 => [[1]]
 => [1] => 1
[1,1] => [1,1]
 => [[1],[2]]
 => [1,1] => 2
[2] => [2]
 => [[1,2]]
 => [2] => 1
[1,1,1] => [1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 3
[1,2] => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[2,1] => [2,1]
 => [[1,3],[2]]
 => [1,2] => 1
[3] => [3]
 => [[1,2,3]]
 => [3] => 1
[1,1,1,1] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 4
[1,1,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 2
[1,2,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 2
[1,3] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[2,1,1] => [2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => 2
[2,2] => [2,2]
 => [[1,2],[3,4]]
 => [2,2] => 2
[3,1] => [3,1]
 => [[1,3,4],[2]]
 => [1,3] => 1
[4] => [4]
 => [[1,2,3,4]]
 => [4] => 1
[1,1,1,1,1] => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 5
[1,1,1,2] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 3
[1,1,2,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 3
[1,1,3] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 2
[1,2,1,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 3
[1,2,2] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 2
[1,3,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 2
[1,4] => [4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
[2,1,1,1] => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [1,1,1,2] => 3
[2,1,2] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 2
[2,2,1] => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => 2
[2,3] => [3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 1
[3,1,1] => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => 2
[3,2] => [3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => 1
[4,1] => [4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => 1
[5] => [5]
 => [[1,2,3,4,5]]
 => [5] => 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 6
[1,1,1,1,2] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 4
[1,1,1,2,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 4
[1,1,1,3] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 3
[1,1,2,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 4
[1,1,2,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 2
[1,1,3,1] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 3
[1,1,4] => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => 2
[1,2,1,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 4
[1,2,1,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 2
[1,2,2,1] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 2
[1,2,3] => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => 1
[1,3,1,1] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [1,1,1,3] => 3
[1,3,2] => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => 1
[1,4,1] => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => 2
[1,5] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => 1
[2,1,1,1,1] => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [1,1,1,1,2] => 4
[2,1,1,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 2
[2,1,2,1] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [1,1,2,2] => 2
Description
The maximal number of repetitions of an integer composition. This is the maximal part of the composition obtained by applying the delta morphism.
Matching statistic: St000771
(load all 3 compositions to match this statistic)
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 56%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,1] => ([(0,1)],2)
 => ([],1)
 => 1
[2] => ([],2)
 => ([],2)
 => ? = 2
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
[1,2] => ([(1,2)],3)
 => ([],2)
 => ? ∊ {1,3}
[2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[3] => ([],3)
 => ([],3)
 => ? ∊ {1,3}
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {2,2,2,4}
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => ? ∊ {2,2,2,4}
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,4}
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[4] => ([],4)
 => ([],4)
 => ? ∊ {2,2,2,4}
[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)
 => ([],1)
 => 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {1,2,2,2,3,3,3,5}
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {1,2,2,2,3,3,3,5}
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,2,2,2,3,3,3,5}
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => ? ∊ {1,2,2,2,3,3,3,5}
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,2,2,2,3,3,3,5}
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => ? ∊ {1,2,2,2,3,3,3,5}
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,3,3,3,5}
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5] => ([],5)
 => ([],5)
 => ? ∊ {1,2,2,2,3,3,3,5}
[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)
 => ([],1)
 => 1
[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)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1)],2)
 => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1)],2)
 => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1)],2)
 => 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,5] => ([(4,5)],6)
 => ([],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1)],2)
 => 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,4] => ([(3,5),(4,5)],6)
 => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[6] => ([],6)
 => ([],6)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],1)
 => 1
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[1,1,2,1,2] => ([(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)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,2,2,1] => ([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,6] => ([(5,6)],7)
 => ([],6)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
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
(load all 2 compositions to match this statistic)
Mapping
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 56%
Values
[1] => ([],1)
 => ([],1)
 => 1
[1,1] => ([(0,1)],2)
 => ([],1)
 => 1
[2] => ([],2)
 => ([],2)
 => ? = 2
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([],1)
 => 1
[1,2] => ([(1,2)],3)
 => ([],2)
 => ? ∊ {1,3}
[2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 1
[3] => ([],3)
 => ([],3)
 => ? ∊ {1,3}
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {2,2,2,4}
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[1,3] => ([(2,3)],4)
 => ([],3)
 => ? ∊ {2,2,2,4}
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 1
[2,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? ∊ {2,2,2,4}
[3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[4] => ([],4)
 => ([],4)
 => ? ∊ {2,2,2,4}
[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)
 => ([],1)
 => 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([],2)
 => ? ∊ {1,2,2,2,3,3,3,5}
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([],3)
 => ? ∊ {1,2,2,2,3,3,3,5}
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,2,2,2,3,3,3,5}
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,4] => ([(3,4)],5)
 => ([],4)
 => ? ∊ {1,2,2,2,3,3,3,5}
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? ∊ {1,2,2,2,3,3,3,5}
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,3] => ([(2,4),(3,4)],5)
 => ([(2,3)],4)
 => ? ∊ {1,2,2,2,3,3,3,5}
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,2,2,2,3,3,3,5}
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[5] => ([],5)
 => ([],5)
 => ? ∊ {1,2,2,2,3,3,3,5}
[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)
 => ([],1)
 => 1
[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)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1)],2)
 => 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1)],2)
 => 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1)],2)
 => 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,5] => ([(4,5)],6)
 => ([],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1)],2)
 => 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,4] => ([(3,5),(4,5)],6)
 => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[6] => ([],6)
 => ([],6)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,3,4,4,4,4,6}
[1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],1)
 => 1
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[1,1,2,1,2] => ([(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)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,2,2,1] => ([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[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)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,6] => ([(5,6)],7)
 => ([],6)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2)],3)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,2),(1,3),(2,3)],4)
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,5,5,5,5,5,7}
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: St001878
(load all 2 compositions to match this statistic)
Mapping
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 22%
Values
[1] => [[1],[]]
 => ([],1)
 => ([],1)
 => ? = 1
[1,1] => [[1,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,2}
[2] => [[2],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,2}
[1,1,1] => [[1,1,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,3}
[1,2] => [[2,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,3}
[2,1] => [[2,2],[1]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,3}
[3] => [[3],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,3}
[1,1,1,1] => [[1,1,1,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,4}
[1,1,2] => [[2,1,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,4}
[1,2,1] => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2,2,2,4}
[1,3] => [[3,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,4}
[2,1,1] => [[2,2,2],[1,1]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,4}
[2,2] => [[3,2],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2,2,2,4}
[3,1] => [[3,3],[2]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,4}
[4] => [[4],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,2,2,2,2,4}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[1,1,1,2] => [[2,1,1,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[1,1,3] => [[3,1,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[1,2,2] => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,1] => [[3,3,1],[2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[1,4] => [[4,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[2,1,2] => [[3,2,2],[1,1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[2,2,1] => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3] => [[4,2],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[3,1,1] => [[3,3,3],[2,2]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[3,2] => [[4,3],[2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[4,1] => [[4,4],[3]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[5] => [[5],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,5}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[1,1,1,3] => [[3,1,1,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,2,2] => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,3,1] => [[3,3,1,1],[2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[1,1,4] => [[4,1,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[1,2,3] => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[1,3,2] => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,4,1] => [[4,4,1],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[1,5] => [[5,1],[]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,3] => [[4,2,2],[1,1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,2,2] => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[2,3,1] => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,4] => [[5,2],[1]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[3,1,2] => [[4,3,3],[2,2]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[3,2,1] => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,3] => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,1,1] => [[4,4,4],[3,3]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[4,2] => [[5,4],[3]]
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[5,1] => [[5,5],[4]]
 => ([],1)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,4,4,4,4,4,6}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,2,3] => [[4,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,3,2] => [[4,3,1,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,1,3] => [[4,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,4] => [[5,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,4,2] => [[5,4,1],[3]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,4,1] => [[5,5,2],[4,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,3] => [[5,3,3],[2,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,4] => [[6,3],[2]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,2,1] => [[5,5,4],[4,3]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,3] => [[6,4],[3]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,1,3,2] => [[4,3,1,1,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[1,4,1,2] => [[5,4,4,1],[3,3]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[2,4,1,1] => [[5,5,5,2],[4,4,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,1,1,3] => [[5,3,3,3],[2,2,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
 => ([(0,2),(2,1)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,1,2,1] => [[5,5,4,4],[4,3,3]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001399
Mapping
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St001399: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 67%
Values
[1] => [1,0]
 => [[]]
 => ([(0,1)],2)
 => 1
[1,1] => [1,0,1,0]
 => [[],[]]
 => ([(0,2),(1,2)],3)
 => 2
[2] => [1,1,0,0]
 => [[[]]]
 => ([(0,2),(2,1)],3)
 => 1
[1,1,1] => [1,0,1,0,1,0]
 => [[],[],[]]
 => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,2] => [1,0,1,1,0,0]
 => [[],[[]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,1] => [1,1,0,0,1,0]
 => [[[]],[]]
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[3] => [1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [[],[],[[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[1,3] => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [[[]],[[]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
[3,1] => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 1
[4] => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[[]]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [[],[],[[]],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [[],[[[]]],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [[],[[[[]]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [[[]],[],[],[]]
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [[[]],[],[[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [[[]],[[]],[]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [[[]],[[[]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [[[[]]],[],[]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [[[[]]],[[]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [[[[[]]]],[]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [[[[[[]]]]]]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[],[[]]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 4
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[],[],[],[[]],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[],[],[],[[[]]]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[],[],[[]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 4
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[],[],[[]],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[],[],[[[]]],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[],[],[[[[]]]]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[],[[]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[],[[]],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[],[[]],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[],[[]],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [[],[[[]]],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 3
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[],[[[]]],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [[],[[[[]]]],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[],[[[[[]]]]]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [[[]],[],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [[[]],[],[],[[]]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[[]],[],[[]],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 2
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [[[]],[],[[[]]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[[]],[[]],[],[]]
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 2
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[[]],[[]],[[]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[[]],[[[]]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [[[]],[[[[]]]]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [[[[]]],[],[],[]]
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 3
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [[[[]]],[],[[]]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[[[]]],[[]],[]]
 => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[[[]]],[[[]]]]
 => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [[[[[]]]],[],[]]
 => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7)
 => 2
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [[[[[]]]],[[]]]
 => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[[[[[]]]]],[]]
 => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [[[[[[[]]]]]]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,3}
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [[],[],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [[],[],[],[],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [[],[],[],[],[[]],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [[],[],[],[],[[[]]]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [[],[],[],[[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [[],[],[],[[]],[[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [[],[],[],[[[]]],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [[],[],[],[[[[]]]]]
 => ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [[],[],[[]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [[],[],[[]],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [[],[],[[]],[[]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [[],[],[[]],[[[]]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [[],[],[[[]]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [[],[],[[[]]],[[]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [[],[],[[[[]]]],[]]
 => ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [[],[],[[[[[]]]]]]
 => ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [[],[[]],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [[],[[]],[],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [[],[[]],[],[[]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [[],[[]],[],[[[]]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [[],[[]],[[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [[],[[]],[[]],[[]]]
 => ([(0,7),(1,6),(2,5),(3,4),(4,7),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [[],[[]],[[[]]],[]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [[],[[]],[[[[]]]]]
 => ([(0,7),(1,6),(2,4),(3,7),(4,7),(5,3),(6,5)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [[],[[[]]],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,5),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [[],[[[]]],[],[[]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [[],[[[]]],[[]],[]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [[],[[[]]],[[[]]]]
 => ([(0,7),(1,6),(2,5),(3,7),(4,7),(5,3),(6,4)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [[],[[[[]]]],[],[]]
 => ([(0,4),(1,7),(2,7),(3,7),(4,6),(5,7),(6,5)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [[],[[[[]]]],[[]]]
 => ([(0,7),(1,6),(2,4),(3,7),(4,7),(5,3),(6,5)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [[],[[[[[]]]]],[]]
 => ([(0,7),(1,7),(2,6),(3,7),(4,5),(5,3),(6,4)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[],[[[[[[]]]]]]]
 => ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [[[]],[],[],[],[],[]]
 => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [[[]],[],[],[],[[]]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [[[]],[],[],[[]],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [[[]],[],[],[[[]]]]
 => ([(0,7),(1,7),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [[[]],[],[[]],[],[]]
 => ([(0,7),(1,7),(2,7),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,7}
Description
The distinguishing number of a poset. This is the minimal number of colours needed to colour the vertices of a poset, such that only the trivial automorphism of the poset preserves the colouring. See also [[St000469]], which is the same concept for graphs.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001118The acyclic chromatic index of a graph. St001597The Frobenius rank of a skew partition. St001235The global dimension of the corresponding Comp-Nakayama algebra.