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Your data matches 23 different statistics following compositions of up to 3 maps.
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Matching statistic: St000904
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(load all 90 compositions to match this statistic)
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
[1,3,1,1] => 3
[1,4,1] => 2
[2,1,1,1,1] => 4
[2,1,1,2] => 2
[2,1,2,1] => 2
[2,2,1,1] => 2
[3,1,1,1] => 3
[4,1,1] => 2
Description
The maximal number of repetitions of an integer composition.
Matching statistic: St001933
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%
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
[1,3,1,1] => [3,1,1,1]
=> 3
[1,4,1] => [4,1,1]
=> 2
[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
[2,2,1,1] => [2,2,1,1]
=> 2
[3,1,1,1] => [3,1,1,1]
=> 3
[4,1,1] => [4,1,1]
=> 2
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000392
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
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%
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
[1,3,1,1] => [3,1,1,1]
=> 1001110 => 3
[1,4,1] => [4,1,1]
=> 1000110 => 2
[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
[2,2,1,1] => [2,2,1,1]
=> 110110 => 2
[3,1,1,1] => [3,1,1,1]
=> 1001110 => 3
[4,1,1] => [4,1,1]
=> 1000110 => 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)
(load all 3 compositions to match this statistic)
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%
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
[1,3,1,1] => [3,1,1,1]
=> 1001110 => 3
[1,4,1] => [4,1,1]
=> 1000110 => 2
[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
[2,2,1,1] => [2,2,1,1]
=> 110110 => 2
[3,1,1,1] => [3,1,1,1]
=> 1001110 => 3
[4,1,1] => [4,1,1]
=> 1000110 => 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: St000686
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000686: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000686: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,1] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[2] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[3] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,3] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,2] => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[4] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,4] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[2,3] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,2] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,1] => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[5] => [1] => [1,0]
=> [1,0]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
Description
The finitistic dominant dimension of a Dyck path.
To every LNakayama algebra there is a corresponding Dyck path, see also [[St000684]]. We associate the finitistic dominant dimension of the algebra to the corresponding Dyck path.
Matching statistic: St000757
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%
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
[1,3,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 3
[1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 2
[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
[2,2,1,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 2
[3,1,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 3
[4,1,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 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)
(load all 2 compositions to match this statistic)
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%
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
[1,3,1,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 3
[1,4,1] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 2
[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
[2,2,1,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 2
[3,1,1,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 3
[4,1,1] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 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: St001399
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001399: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001399: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => ([],1)
=> 1
[1,1] => [1,0,1,0]
=> [2,1] => ([],2)
=> 2
[2] => [1,1,0,0]
=> [1,2] => ([(0,1)],2)
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => ([],3)
=> 3
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 1
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => ([(1,2)],3)
=> 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => ([],4)
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => ([(2,3)],4)
=> 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => ([(2,3)],4)
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => ([(2,3)],4)
=> 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => ([],5)
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => ([(3,4)],5)
=> 3
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => ([(3,4)],5)
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => ([(3,4)],5)
=> 3
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => ([],6)
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => ([(4,5)],6)
=> 4
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => ([(4,5)],6)
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => ([(3,4),(4,5)],6)
=> 3
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => ([(4,5)],6)
=> 4
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => ([(2,5),(3,4)],6)
=> 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => ([(3,4),(4,5)],6)
=> 3
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ([(2,3),(3,5),(5,4)],6)
=> 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => ([(4,5)],6)
=> 4
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => ([(2,5),(3,4)],6)
=> 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => ([(2,5),(3,4)],6)
=> 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => ([(3,4),(4,5)],6)
=> 3
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => ([(2,3),(3,5),(5,4)],6)
=> 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => ([(4,5)],6)
=> 4
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => ([(2,5),(3,4)],6)
=> 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => ([(2,5),(3,4)],6)
=> 2
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
=> 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
=> 3
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
=> 2
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.
Matching statistic: St001235
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Values
[1] => [1]
=> 10 => [1,2] => 2 = 1 + 1
[1,1] => [1,1]
=> 110 => [1,1,2] => 3 = 2 + 1
[2] => [2]
=> 100 => [1,3] => 2 = 1 + 1
[1,1,1] => [1,1,1]
=> 1110 => [1,1,1,2] => 4 = 3 + 1
[1,2] => [2,1]
=> 1010 => [1,2,2] => 2 = 1 + 1
[2,1] => [2,1]
=> 1010 => [1,2,2] => 2 = 1 + 1
[3] => [3]
=> 1000 => [1,4] => 2 = 1 + 1
[1,1,1,1] => [1,1,1,1]
=> 11110 => [1,1,1,1,2] => 5 = 4 + 1
[1,1,2] => [2,1,1]
=> 10110 => [1,2,1,2] => 3 = 2 + 1
[1,2,1] => [2,1,1]
=> 10110 => [1,2,1,2] => 3 = 2 + 1
[1,3] => [3,1]
=> 10010 => [1,3,2] => 2 = 1 + 1
[2,1,1] => [2,1,1]
=> 10110 => [1,2,1,2] => 3 = 2 + 1
[2,2] => [2,2]
=> 1100 => [1,1,3] => 3 = 2 + 1
[3,1] => [3,1]
=> 10010 => [1,3,2] => 2 = 1 + 1
[4] => [4]
=> 10000 => [1,5] => 2 = 1 + 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => ? = 5 + 1
[1,1,1,2] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 3 + 1
[1,1,2,1] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 3 + 1
[1,1,3] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 2 + 1
[1,2,1,1] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 3 + 1
[1,2,2] => [2,2,1]
=> 11010 => [1,1,2,2] => 3 = 2 + 1
[1,3,1] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 2 + 1
[1,4] => [4,1]
=> 100010 => [1,4,2] => ? = 1 + 1
[2,1,1,1] => [2,1,1,1]
=> 101110 => [1,2,1,1,2] => ? = 3 + 1
[2,1,2] => [2,2,1]
=> 11010 => [1,1,2,2] => 3 = 2 + 1
[2,2,1] => [2,2,1]
=> 11010 => [1,1,2,2] => 3 = 2 + 1
[2,3] => [3,2]
=> 10100 => [1,2,3] => 2 = 1 + 1
[3,1,1] => [3,1,1]
=> 100110 => [1,3,1,2] => ? = 2 + 1
[3,2] => [3,2]
=> 10100 => [1,2,3] => 2 = 1 + 1
[4,1] => [4,1]
=> 100010 => [1,4,2] => ? = 1 + 1
[5] => [5]
=> 100000 => [1,6] => ? = 1 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => ? = 6 + 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => ? = 4 + 1
[1,1,1,2,1] => [2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => ? = 4 + 1
[1,1,1,3] => [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 3 + 1
[1,1,2,1,1] => [2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => ? = 4 + 1
[1,1,2,2] => [2,2,1,1]
=> 110110 => [1,1,2,1,2] => ? = 2 + 1
[1,1,3,1] => [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 3 + 1
[1,1,4] => [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 2 + 1
[1,2,1,1,1] => [2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => ? = 4 + 1
[1,2,1,2] => [2,2,1,1]
=> 110110 => [1,1,2,1,2] => ? = 2 + 1
[1,2,2,1] => [2,2,1,1]
=> 110110 => [1,1,2,1,2] => ? = 2 + 1
[1,3,1,1] => [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 3 + 1
[1,4,1] => [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 2 + 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => ? = 4 + 1
[2,1,1,2] => [2,2,1,1]
=> 110110 => [1,1,2,1,2] => ? = 2 + 1
[2,1,2,1] => [2,2,1,1]
=> 110110 => [1,1,2,1,2] => ? = 2 + 1
[2,2,1,1] => [2,2,1,1]
=> 110110 => [1,1,2,1,2] => ? = 2 + 1
[3,1,1,1] => [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 3 + 1
[4,1,1] => [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 2 + 1
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St000907
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000907: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00262: Binary words —poset of factors⟶ Posets
St000907: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => ([(0,1)],2)
=> 2 = 1 + 1
[1,1] => 11 => ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1] => 111 => ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[1,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[2,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,1,1,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,1,2] => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 2 + 1
[1,2,1] => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2 + 1
[1,3] => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[2,1,1] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 2 + 1
[2,2] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 2 + 1
[3,1] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[1,1,1,1,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[1,1,1,2] => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 3 + 1
[1,1,2,1] => 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3 + 1
[1,1,3] => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 2 + 1
[1,2,1,1] => 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 3 + 1
[1,2,2] => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 2 + 1
[1,3,1] => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 + 1
[1,4] => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[2,1,1,1] => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 3 + 1
[2,1,2] => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 2 + 1
[2,2,1] => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 2 + 1
[2,3] => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 1 + 1
[3,1,1] => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2 + 1
[3,2] => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
[4,1] => 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 + 1
[5] => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 1 + 1
[1,1,1,1,1,1] => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 6 + 1
[1,1,1,1,2] => 111110 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 4 + 1
[1,1,1,2,1] => 111101 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 4 + 1
[1,1,1,3] => 111100 => ([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> ? = 3 + 1
[1,1,2,1,1] => 111011 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 4 + 1
[1,1,2,2] => 111010 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 2 + 1
[1,1,3,1] => 111001 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 3 + 1
[1,1,4] => 111000 => ([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> ? = 2 + 1
[1,2,1,1,1] => 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 4 + 1
[1,2,1,2] => 110110 => ([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> ? = 2 + 1
[1,2,2,1] => 110101 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 2 + 1
[1,3,1,1] => 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 3 + 1
[1,4,1] => 110001 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 2 + 1
[2,1,1,1,1] => 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 4 + 1
[2,1,1,2] => 101110 => ([(0,3),(0,4),(1,2),(1,11),(1,15),(2,7),(2,12),(3,13),(3,14),(4,1),(4,13),(4,14),(6,9),(7,10),(8,6),(9,5),(10,5),(11,7),(12,9),(12,10),(13,8),(13,15),(14,8),(14,11),(15,6),(15,12)],16)
=> ? = 2 + 1
[2,1,2,1] => 101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 2 + 1
[2,2,1,1] => 101011 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 2 + 1
[3,1,1,1] => 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 3 + 1
[4,1,1] => 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 2 + 1
Description
The number of maximal antichains of minimal length in a poset.
The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St000650The number of 3-rises of a permutation. St001060The distinguishing index of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.
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