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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000904
(load all 91 compositions to match this statistic)
(load all 91 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
[1,2,3] => 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
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
[1,2,3] => [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)
(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
[1,2,3] => [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)
(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
[1,2,3] => [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
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
[1,2,3] => [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)
(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
[1,2,3] => [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: St001399
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%
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
[1,3,1,1] => [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
[1,4,1] => [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
[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
[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)
=> ? = 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
[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
[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
[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
[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)
=> ? = 2
[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
[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
[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] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 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)
=> ? = 5
[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)
=> ? = 5
[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)
=> ? = 4
[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)
=> ? = 5
[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)
=> ? = 3
[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)
=> ? = 4
[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)
=> ? = 3
[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)
=> ? = 5
[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)
=> ? = 3
[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)
=> ? = 3
[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)
=> ? = 2
[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)
=> ? = 4
[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)
=> ? = 2
[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)
=> ? = 3
[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)
=> ? = 2
[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)
=> ? = 5
[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)
=> ? = 3
[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)
=> ? = 3
[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)
=> ? = 2
[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)
=> ? = 3
[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)
=> ? = 3
[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)
=> ? = 2
[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
[1,3,1,1,1] => [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)
=> ? = 4
[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)
=> ? = 2
[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)
=> ? = 2
[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)
=> ? = 2
[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)
=> ? = 3
[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
[1,5,1] => [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)
=> ? = 2
[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
[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)
=> ? = 5
[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)
=> ? = 3
[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)
=> ? = 3
[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)
=> ? = 2
[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)
=> ? = 3
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: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 44%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 44%
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
[1,2,3] => [3,2,1]
=> 101010 => [1,2,2,2] => ? = 1 + 1
[1,3,1,1] => [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 3 + 1
[1,3,2] => [3,2,1]
=> 101010 => [1,2,2,2] => ? = 1 + 1
[1,4,1] => [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 2 + 1
[1,5] => [5,1]
=> 1000010 => [1,5,2] => ? = 1 + 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
[2,1,3] => [3,2,1]
=> 101010 => [1,2,2,2] => ? = 1 + 1
[2,2,1,1] => [2,2,1,1]
=> 110110 => [1,1,2,1,2] => ? = 2 + 1
[2,2,2] => [2,2,2]
=> 11100 => [1,1,1,3] => 4 = 3 + 1
[2,3,1] => [3,2,1]
=> 101010 => [1,2,2,2] => ? = 1 + 1
[2,4] => [4,2]
=> 100100 => [1,3,3] => ? = 1 + 1
[3,1,1,1] => [3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ? = 3 + 1
[3,1,2] => [3,2,1]
=> 101010 => [1,2,2,2] => ? = 1 + 1
[3,2,1] => [3,2,1]
=> 101010 => [1,2,2,2] => ? = 1 + 1
[3,3] => [3,3]
=> 11000 => [1,1,4] => 3 = 2 + 1
[4,1,1] => [4,1,1]
=> 1000110 => [1,4,1,2] => ? = 2 + 1
[4,2] => [4,2]
=> 100100 => [1,3,3] => ? = 1 + 1
[5,1] => [5,1]
=> 1000010 => [1,5,2] => ? = 1 + 1
[6] => [6]
=> 1000000 => [1,7] => ? = 1 + 1
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => ? = 7 + 1
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => ? = 5 + 1
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => ? = 5 + 1
[1,1,1,1,3] => [3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => ? = 4 + 1
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => ? = 5 + 1
[1,1,1,2,2] => [2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => ? = 3 + 1
[1,1,1,3,1] => [3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => ? = 4 + 1
[1,1,1,4] => [4,1,1,1]
=> 10001110 => [1,4,1,1,2] => ? = 3 + 1
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => ? = 5 + 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".
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