Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St000148
(load all 2 compositions to match this statistic)
Mapping
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 0 = 1 - 1
[1,1]
 => 2 = 3 - 1
[3]
 => 1 = 2 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 3 = 4 - 1
[4]
 => 0 = 1 - 1
[3,1]
 => 2 = 3 - 1
[2,2]
 => 0 = 1 - 1
[2,1,1]
 => 2 = 3 - 1
[1,1,1,1]
 => 4 = 5 - 1
[5]
 => 1 = 2 - 1
[4,1]
 => 1 = 2 - 1
[3,2]
 => 1 = 2 - 1
[3,1,1]
 => 3 = 4 - 1
[2,2,1]
 => 1 = 2 - 1
[2,1,1,1]
 => 3 = 4 - 1
[1,1,1,1,1]
 => 5 = 6 - 1
[6]
 => 0 = 1 - 1
[5,1]
 => 2 = 3 - 1
[4,2]
 => 0 = 1 - 1
[4,1,1]
 => 2 = 3 - 1
[3,3]
 => 2 = 3 - 1
[3,2,1]
 => 2 = 3 - 1
[3,1,1,1]
 => 4 = 5 - 1
[2,2,2]
 => 0 = 1 - 1
[2,2,1,1]
 => 2 = 3 - 1
[2,1,1,1,1]
 => 4 = 5 - 1
[1,1,1,1,1,1]
 => 6 = 7 - 1
Description
The number of odd parts of a partition.
Matching statistic: St000992
Mapping
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 = 2 - 1
[2]
 => 2 = 3 - 1
[1,1]
 => 0 = 1 - 1
[3]
 => 3 = 4 - 1
[2,1]
 => 1 = 2 - 1
[1,1,1]
 => 1 = 2 - 1
[4]
 => 4 = 5 - 1
[3,1]
 => 2 = 3 - 1
[2,2]
 => 0 = 1 - 1
[2,1,1]
 => 2 = 3 - 1
[1,1,1,1]
 => 0 = 1 - 1
[5]
 => 5 = 6 - 1
[4,1]
 => 3 = 4 - 1
[3,2]
 => 1 = 2 - 1
[3,1,1]
 => 3 = 4 - 1
[2,2,1]
 => 1 = 2 - 1
[2,1,1,1]
 => 1 = 2 - 1
[1,1,1,1,1]
 => 1 = 2 - 1
[6]
 => 6 = 7 - 1
[5,1]
 => 4 = 5 - 1
[4,2]
 => 2 = 3 - 1
[4,1,1]
 => 4 = 5 - 1
[3,3]
 => 0 = 1 - 1
[3,2,1]
 => 2 = 3 - 1
[3,1,1,1]
 => 2 = 3 - 1
[2,2,2]
 => 2 = 3 - 1
[2,2,1,1]
 => 0 = 1 - 1
[2,1,1,1,1]
 => 2 = 3 - 1
[1,1,1,1,1,1]
 => 0 = 1 - 1
Description
The alternating sum of the parts of an integer partition. For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St000288
(load all 19 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 = 2 - 1
[2]
 => 0 => 0 = 1 - 1
[1,1]
 => 11 => 2 = 3 - 1
[3]
 => 1 => 1 = 2 - 1
[2,1]
 => 01 => 1 = 2 - 1
[1,1,1]
 => 111 => 3 = 4 - 1
[4]
 => 0 => 0 = 1 - 1
[3,1]
 => 11 => 2 = 3 - 1
[2,2]
 => 00 => 0 = 1 - 1
[2,1,1]
 => 011 => 2 = 3 - 1
[1,1,1,1]
 => 1111 => 4 = 5 - 1
[5]
 => 1 => 1 = 2 - 1
[4,1]
 => 01 => 1 = 2 - 1
[3,2]
 => 10 => 1 = 2 - 1
[3,1,1]
 => 111 => 3 = 4 - 1
[2,2,1]
 => 001 => 1 = 2 - 1
[2,1,1,1]
 => 0111 => 3 = 4 - 1
[1,1,1,1,1]
 => 11111 => 5 = 6 - 1
[6]
 => 0 => 0 = 1 - 1
[5,1]
 => 11 => 2 = 3 - 1
[4,2]
 => 00 => 0 = 1 - 1
[4,1,1]
 => 011 => 2 = 3 - 1
[3,3]
 => 11 => 2 = 3 - 1
[3,2,1]
 => 101 => 2 = 3 - 1
[3,1,1,1]
 => 1111 => 4 = 5 - 1
[2,2,2]
 => 000 => 0 = 1 - 1
[2,2,1,1]
 => 0011 => 2 = 3 - 1
[2,1,1,1,1]
 => 01111 => 4 = 5 - 1
[1,1,1,1,1,1]
 => 111111 => 6 = 7 - 1
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St001372
(load all 14 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 = 2 - 1
[2]
 => 0 => 0 = 1 - 1
[1,1]
 => 11 => 2 = 3 - 1
[3]
 => 1 => 1 = 2 - 1
[2,1]
 => 01 => 1 = 2 - 1
[1,1,1]
 => 111 => 3 = 4 - 1
[4]
 => 0 => 0 = 1 - 1
[3,1]
 => 11 => 2 = 3 - 1
[2,2]
 => 00 => 0 = 1 - 1
[2,1,1]
 => 011 => 2 = 3 - 1
[1,1,1,1]
 => 1111 => 4 = 5 - 1
[5]
 => 1 => 1 = 2 - 1
[4,1]
 => 01 => 1 = 2 - 1
[3,2]
 => 10 => 1 = 2 - 1
[3,1,1]
 => 111 => 3 = 4 - 1
[2,2,1]
 => 001 => 1 = 2 - 1
[2,1,1,1]
 => 0111 => 3 = 4 - 1
[1,1,1,1,1]
 => 11111 => 5 = 6 - 1
[6]
 => 0 => 0 = 1 - 1
[5,1]
 => 11 => 2 = 3 - 1
[4,2]
 => 00 => 0 = 1 - 1
[4,1,1]
 => 011 => 2 = 3 - 1
[3,3]
 => 11 => 2 = 3 - 1
[3,2,1]
 => 101 => 2 = 3 - 1
[3,1,1,1]
 => 1111 => 4 = 5 - 1
[2,2,2]
 => 000 => 0 = 1 - 1
[2,2,1,1]
 => 0011 => 2 = 3 - 1
[2,1,1,1,1]
 => 01111 => 4 = 5 - 1
[1,1,1,1,1,1]
 => 111111 => 6 = 7 - 1
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: St000022
(load all 5 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => [1] => 1 = 2 - 1
[2]
 => [[1,2]]
 => [1,2] => 2 = 3 - 1
[1,1]
 => [[1],[2]]
 => [2,1] => 0 = 1 - 1
[3]
 => [[1,2,3]]
 => [1,2,3] => 3 = 4 - 1
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1 = 2 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1 = 2 - 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 4 = 5 - 1
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2 = 3 - 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 0 = 1 - 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2 = 3 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0 = 1 - 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => 5 = 6 - 1
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => 3 = 4 - 1
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => 1 = 2 - 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 3 = 4 - 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 1 = 2 - 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1 = 2 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 1 = 2 - 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 6 = 7 - 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 4 = 5 - 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => 2 = 3 - 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => 4 = 5 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 0 = 1 - 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => 2 = 3 - 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => 2 = 3 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => 2 = 3 - 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => 0 = 1 - 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => 2 = 3 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 0 = 1 - 1
Description
The number of fixed points of a permutation.
Matching statistic: St000392
(load all 6 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 => 1 = 2 - 1
[2]
 => 0 => 0 => 0 = 1 - 1
[1,1]
 => 11 => 11 => 2 = 3 - 1
[3]
 => 1 => 1 => 1 = 2 - 1
[2,1]
 => 01 => 01 => 1 = 2 - 1
[1,1,1]
 => 111 => 111 => 3 = 4 - 1
[4]
 => 0 => 0 => 0 = 1 - 1
[3,1]
 => 11 => 11 => 2 = 3 - 1
[2,2]
 => 00 => 00 => 0 = 1 - 1
[2,1,1]
 => 011 => 011 => 2 = 3 - 1
[1,1,1,1]
 => 1111 => 1111 => 4 = 5 - 1
[5]
 => 1 => 1 => 1 = 2 - 1
[4,1]
 => 01 => 01 => 1 = 2 - 1
[3,2]
 => 10 => 01 => 1 = 2 - 1
[3,1,1]
 => 111 => 111 => 3 = 4 - 1
[2,2,1]
 => 001 => 001 => 1 = 2 - 1
[2,1,1,1]
 => 0111 => 0111 => 3 = 4 - 1
[1,1,1,1,1]
 => 11111 => 11111 => 5 = 6 - 1
[6]
 => 0 => 0 => 0 = 1 - 1
[5,1]
 => 11 => 11 => 2 = 3 - 1
[4,2]
 => 00 => 00 => 0 = 1 - 1
[4,1,1]
 => 011 => 011 => 2 = 3 - 1
[3,3]
 => 11 => 11 => 2 = 3 - 1
[3,2,1]
 => 101 => 011 => 2 = 3 - 1
[3,1,1,1]
 => 1111 => 1111 => 4 = 5 - 1
[2,2,2]
 => 000 => 000 => 0 = 1 - 1
[2,2,1,1]
 => 0011 => 0011 => 2 = 3 - 1
[2,1,1,1,1]
 => 01111 => 01111 => 4 = 5 - 1
[1,1,1,1,1,1]
 => 111111 => 111111 => 6 = 7 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001419
(load all 6 compositions to match this statistic)
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 1 => 1 = 2 - 1
[2]
 => 0 => 0 => 0 = 1 - 1
[1,1]
 => 11 => 11 => 2 = 3 - 1
[3]
 => 1 => 1 => 1 = 2 - 1
[2,1]
 => 01 => 01 => 1 = 2 - 1
[1,1,1]
 => 111 => 111 => 3 = 4 - 1
[4]
 => 0 => 0 => 0 = 1 - 1
[3,1]
 => 11 => 11 => 2 = 3 - 1
[2,2]
 => 00 => 00 => 0 = 1 - 1
[2,1,1]
 => 011 => 011 => 2 = 3 - 1
[1,1,1,1]
 => 1111 => 1111 => 4 = 5 - 1
[5]
 => 1 => 1 => 1 = 2 - 1
[4,1]
 => 01 => 01 => 1 = 2 - 1
[3,2]
 => 10 => 01 => 1 = 2 - 1
[3,1,1]
 => 111 => 111 => 3 = 4 - 1
[2,2,1]
 => 001 => 001 => 1 = 2 - 1
[2,1,1,1]
 => 0111 => 0111 => 3 = 4 - 1
[1,1,1,1,1]
 => 11111 => 11111 => 5 = 6 - 1
[6]
 => 0 => 0 => 0 = 1 - 1
[5,1]
 => 11 => 11 => 2 = 3 - 1
[4,2]
 => 00 => 00 => 0 = 1 - 1
[4,1,1]
 => 011 => 011 => 2 = 3 - 1
[3,3]
 => 11 => 11 => 2 = 3 - 1
[3,2,1]
 => 101 => 011 => 2 = 3 - 1
[3,1,1,1]
 => 1111 => 1111 => 4 = 5 - 1
[2,2,2]
 => 000 => 000 => 0 = 1 - 1
[2,2,1,1]
 => 0011 => 0011 => 2 = 3 - 1
[2,1,1,1,1]
 => 01111 => 01111 => 4 = 5 - 1
[1,1,1,1,1,1]
 => 111111 => 111111 => 6 = 7 - 1
Description
The length of the longest palindromic factor beginning with a one of a binary word.
Matching statistic: St000010
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => [1,1]
 => 2
[2]
 => 0 => [2] => [2]
 => 1
[1,1]
 => 11 => [1,1,1] => [1,1,1]
 => 3
[3]
 => 1 => [1,1] => [1,1]
 => 2
[2,1]
 => 01 => [2,1] => [2,1]
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 4
[4]
 => 0 => [2] => [2]
 => 1
[3,1]
 => 11 => [1,1,1] => [1,1,1]
 => 3
[2,2]
 => 00 => [3] => [3]
 => 1
[2,1,1]
 => 011 => [2,1,1] => [2,1,1]
 => 3
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 5
[5]
 => 1 => [1,1] => [1,1]
 => 2
[4,1]
 => 01 => [2,1] => [2,1]
 => 2
[3,2]
 => 10 => [1,2] => [2,1]
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 4
[2,2,1]
 => 001 => [3,1] => [3,1]
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => [2,1,1,1]
 => 4
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 6
[6]
 => 0 => [2] => [2]
 => 1
[5,1]
 => 11 => [1,1,1] => [1,1,1]
 => 3
[4,2]
 => 00 => [3] => [3]
 => 1
[4,1,1]
 => 011 => [2,1,1] => [2,1,1]
 => 3
[3,3]
 => 11 => [1,1,1] => [1,1,1]
 => 3
[3,2,1]
 => 101 => [1,2,1] => [2,1,1]
 => 3
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 5
[2,2,2]
 => 000 => [4] => [4]
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => [3,1,1]
 => 3
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => 5
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
 => 7
Description
The length of the partition.
Matching statistic: St000097
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2]
 => 0 => [2] => ([],2)
 => 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4]
 => 0 => [2] => ([],2)
 => 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,2]
 => 00 => [3] => ([],3)
 => 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1]
 => 1111 => [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)
 => 5
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1]
 => 11111 => [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)
 => 6
[6]
 => 0 => [2] => ([],2)
 => 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4,2]
 => 00 => [3] => ([],3)
 => 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => 1111 => [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)
 => 5
[2,2,2]
 => 000 => [4] => ([],4)
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1]
 => 01111 => [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)
 => 5
[1,1,1,1,1,1]
 => 111111 => [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)
 => 7
Description
The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000098: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2]
 => 0 => [2] => ([],2)
 => 1
[1,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[2,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[4]
 => 0 => [2] => ([],2)
 => 1
[3,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,2]
 => 00 => [3] => ([],3)
 => 1
[2,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,1,1,1]
 => 1111 => [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)
 => 5
[5]
 => 1 => [1,1] => ([(0,1)],2)
 => 2
[4,1]
 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,2]
 => 10 => [1,2] => ([(1,2)],3)
 => 2
[3,1,1]
 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[2,2,1]
 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[2,1,1,1]
 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
[1,1,1,1,1]
 => 11111 => [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)
 => 6
[6]
 => 0 => [2] => ([],2)
 => 1
[5,1]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[4,2]
 => 00 => [3] => ([],3)
 => 1
[4,1,1]
 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,3]
 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[3,2,1]
 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[3,1,1,1]
 => 1111 => [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)
 => 5
[2,2,2]
 => 000 => [4] => ([],4)
 => 1
[2,2,1,1]
 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[2,1,1,1,1]
 => 01111 => [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)
 => 5
[1,1,1,1,1,1]
 => 111111 => [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)
 => 7
Description
The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000172The Grundy number of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000696The number of cycles in the breakpoint graph of a permutation. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St000241The number of cyclical small excedances. St000272The treewidth of a graph. St000297The number of leading ones in a binary word. St000362The size of a minimal vertex cover of a graph. St000475The number of parts equal to 1 in a partition. St000536The pathwidth of a graph. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000895The number of ones on the main diagonal of an alternating sign matrix. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St000822The Hadwiger number of the graph. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000247The number of singleton blocks of a set partition. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001812The biclique partition number of a graph. St000894The trace of an alternating sign matrix. St001330The hat guessing number of a graph. St000806The semiperimeter of the associated bargraph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001903The number of fixed points of a parking function. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000884The number of isolated descents of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000259The diameter of a connected graph. St001645The pebbling number of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000090The variation of a composition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000850The number of 1/2-balanced pairs in a poset.