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Your data matches 38 different statistics following compositions of up to 3 maps.
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Matching statistic: St000148
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 0
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 3
[4]
=> 0
[3,1]
=> 2
[2,2]
=> 0
[2,1,1]
=> 2
[1,1,1,1]
=> 4
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 3
[2,2,1]
=> 1
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 5
[6]
=> 0
[5,1]
=> 2
[4,2]
=> 0
[4,1,1]
=> 2
[3,3]
=> 2
[3,2,1]
=> 2
[3,1,1,1]
=> 4
[2,2,2]
=> 0
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 6
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 3
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 3
[3,3,1]
=> 3
[3,2,2]
=> 1
[3,2,1,1]
=> 3
[3,1,1,1,1]
=> 5
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 7
[8]
=> 0
[7,1]
=> 2
[6,2]
=> 0
[6,1,1]
=> 2
[5,3]
=> 2
[5,2,1]
=> 2
Description
The number of odd parts of a partition.
Matching statistic: St000992
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 0
[3]
=> 3
[2,1]
=> 1
[1,1,1]
=> 1
[4]
=> 4
[3,1]
=> 2
[2,2]
=> 0
[2,1,1]
=> 2
[1,1,1,1]
=> 0
[5]
=> 5
[4,1]
=> 3
[3,2]
=> 1
[3,1,1]
=> 3
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 1
[6]
=> 6
[5,1]
=> 4
[4,2]
=> 2
[4,1,1]
=> 4
[3,3]
=> 0
[3,2,1]
=> 2
[3,1,1,1]
=> 2
[2,2,2]
=> 2
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 0
[7]
=> 7
[6,1]
=> 5
[5,2]
=> 3
[5,1,1]
=> 5
[4,3]
=> 1
[4,2,1]
=> 3
[4,1,1,1]
=> 3
[3,3,1]
=> 1
[3,2,2]
=> 3
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 1
[8]
=> 8
[7,1]
=> 6
[6,2]
=> 4
[6,1,1]
=> 6
[5,3]
=> 2
[5,2,1]
=> 4
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: St000010
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%
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 = 1 + 1
[2]
=> 0 => [2] => [2]
=> 1 = 0 + 1
[1,1]
=> 11 => [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[3]
=> 1 => [1,1] => [1,1]
=> 2 = 1 + 1
[2,1]
=> 01 => [2,1] => [2,1]
=> 2 = 1 + 1
[1,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 4 = 3 + 1
[4]
=> 0 => [2] => [2]
=> 1 = 0 + 1
[3,1]
=> 11 => [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[2,2]
=> 00 => [3] => [3]
=> 1 = 0 + 1
[2,1,1]
=> 011 => [2,1,1] => [2,1,1]
=> 3 = 2 + 1
[1,1,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5 = 4 + 1
[5]
=> 1 => [1,1] => [1,1]
=> 2 = 1 + 1
[4,1]
=> 01 => [2,1] => [2,1]
=> 2 = 1 + 1
[3,2]
=> 10 => [1,2] => [2,1]
=> 2 = 1 + 1
[3,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 4 = 3 + 1
[2,2,1]
=> 001 => [3,1] => [3,1]
=> 2 = 1 + 1
[2,1,1,1]
=> 0111 => [2,1,1,1] => [2,1,1,1]
=> 4 = 3 + 1
[1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6 = 5 + 1
[6]
=> 0 => [2] => [2]
=> 1 = 0 + 1
[5,1]
=> 11 => [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[4,2]
=> 00 => [3] => [3]
=> 1 = 0 + 1
[4,1,1]
=> 011 => [2,1,1] => [2,1,1]
=> 3 = 2 + 1
[3,3]
=> 11 => [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[3,2,1]
=> 101 => [1,2,1] => [2,1,1]
=> 3 = 2 + 1
[3,1,1,1]
=> 1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 5 = 4 + 1
[2,2,2]
=> 000 => [4] => [4]
=> 1 = 0 + 1
[2,2,1,1]
=> 0011 => [3,1,1] => [3,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> 5 = 4 + 1
[1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> 7 = 6 + 1
[7]
=> 1 => [1,1] => [1,1]
=> 2 = 1 + 1
[6,1]
=> 01 => [2,1] => [2,1]
=> 2 = 1 + 1
[5,2]
=> 10 => [1,2] => [2,1]
=> 2 = 1 + 1
[5,1,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 4 = 3 + 1
[4,3]
=> 01 => [2,1] => [2,1]
=> 2 = 1 + 1
[4,2,1]
=> 001 => [3,1] => [3,1]
=> 2 = 1 + 1
[4,1,1,1]
=> 0111 => [2,1,1,1] => [2,1,1,1]
=> 4 = 3 + 1
[3,3,1]
=> 111 => [1,1,1,1] => [1,1,1,1]
=> 4 = 3 + 1
[3,2,2]
=> 100 => [1,3] => [3,1]
=> 2 = 1 + 1
[3,2,1,1]
=> 1011 => [1,2,1,1] => [2,1,1,1]
=> 4 = 3 + 1
[3,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6 = 5 + 1
[2,2,2,1]
=> 0001 => [4,1] => [4,1]
=> 2 = 1 + 1
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => [3,1,1,1]
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> 011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1]
=> 6 = 5 + 1
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> 8 = 7 + 1
[8]
=> 0 => [2] => [2]
=> 1 = 0 + 1
[7,1]
=> 11 => [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[6,2]
=> 00 => [3] => [3]
=> 1 = 0 + 1
[6,1,1]
=> 011 => [2,1,1] => [2,1,1]
=> 3 = 2 + 1
[5,3]
=> 11 => [1,1,1] => [1,1,1]
=> 3 = 2 + 1
[5,2,1]
=> 101 => [1,2,1] => [2,1,1]
=> 3 = 2 + 1
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => [3,1,1,1,1,1,1,1,1] => ?
=> ? = 8 + 1
Description
The length of the partition.
Matching statistic: St000877
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Values
[1]
=> 1 => 0 => 0 => 1
[2]
=> 0 => 1 => 1 => 0
[1,1]
=> 11 => 00 => 00 => 2
[3]
=> 1 => 0 => 0 => 1
[2,1]
=> 01 => 10 => 01 => 1
[1,1,1]
=> 111 => 000 => 000 => 3
[4]
=> 0 => 1 => 1 => 0
[3,1]
=> 11 => 00 => 00 => 2
[2,2]
=> 00 => 11 => 11 => 0
[2,1,1]
=> 011 => 100 => 001 => 2
[1,1,1,1]
=> 1111 => 0000 => 0000 => 4
[5]
=> 1 => 0 => 0 => 1
[4,1]
=> 01 => 10 => 01 => 1
[3,2]
=> 10 => 01 => 01 => 1
[3,1,1]
=> 111 => 000 => 000 => 3
[2,2,1]
=> 001 => 110 => 011 => 1
[2,1,1,1]
=> 0111 => 1000 => 0001 => 3
[1,1,1,1,1]
=> 11111 => 00000 => 00000 => 5
[6]
=> 0 => 1 => 1 => 0
[5,1]
=> 11 => 00 => 00 => 2
[4,2]
=> 00 => 11 => 11 => 0
[4,1,1]
=> 011 => 100 => 001 => 2
[3,3]
=> 11 => 00 => 00 => 2
[3,2,1]
=> 101 => 010 => 001 => 2
[3,1,1,1]
=> 1111 => 0000 => 0000 => 4
[2,2,2]
=> 000 => 111 => 111 => 0
[2,2,1,1]
=> 0011 => 1100 => 0011 => 2
[2,1,1,1,1]
=> 01111 => 10000 => 00001 => 4
[1,1,1,1,1,1]
=> 111111 => 000000 => 000000 => 6
[7]
=> 1 => 0 => 0 => 1
[6,1]
=> 01 => 10 => 01 => 1
[5,2]
=> 10 => 01 => 01 => 1
[5,1,1]
=> 111 => 000 => 000 => 3
[4,3]
=> 01 => 10 => 01 => 1
[4,2,1]
=> 001 => 110 => 011 => 1
[4,1,1,1]
=> 0111 => 1000 => 0001 => 3
[3,3,1]
=> 111 => 000 => 000 => 3
[3,2,2]
=> 100 => 011 => 011 => 1
[3,2,1,1]
=> 1011 => 0100 => 0001 => 3
[3,1,1,1,1]
=> 11111 => 00000 => 00000 => 5
[2,2,2,1]
=> 0001 => 1110 => 0111 => 1
[2,2,1,1,1]
=> 00111 => 11000 => 00011 => 3
[2,1,1,1,1,1]
=> 011111 => 100000 => 000001 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => 0000000 => 7
[8]
=> 0 => 1 => 1 => 0
[7,1]
=> 11 => 00 => 00 => 2
[6,2]
=> 00 => 11 => 11 => 0
[6,1,1]
=> 011 => 100 => 001 => 2
[5,3]
=> 11 => 00 => 00 => 2
[5,2,1]
=> 101 => 010 => 001 => 2
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 00000000000 => 00000000000 => ? = 11
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 1100000000 => ? => ? ∊ {8,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 000000000000 => 000000000000 => ? ∊ {8,12}
Description
The depth of the binary word interpreted as a path.
This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2].
The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St000326
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Values
[1]
=> 1 => 0 => 0 => 2 = 1 + 1
[2]
=> 0 => 1 => 1 => 1 = 0 + 1
[1,1]
=> 11 => 00 => 00 => 3 = 2 + 1
[3]
=> 1 => 0 => 0 => 2 = 1 + 1
[2,1]
=> 01 => 10 => 01 => 2 = 1 + 1
[1,1,1]
=> 111 => 000 => 000 => 4 = 3 + 1
[4]
=> 0 => 1 => 1 => 1 = 0 + 1
[3,1]
=> 11 => 00 => 00 => 3 = 2 + 1
[2,2]
=> 00 => 11 => 11 => 1 = 0 + 1
[2,1,1]
=> 011 => 100 => 001 => 3 = 2 + 1
[1,1,1,1]
=> 1111 => 0000 => 0000 => 5 = 4 + 1
[5]
=> 1 => 0 => 0 => 2 = 1 + 1
[4,1]
=> 01 => 10 => 01 => 2 = 1 + 1
[3,2]
=> 10 => 01 => 01 => 2 = 1 + 1
[3,1,1]
=> 111 => 000 => 000 => 4 = 3 + 1
[2,2,1]
=> 001 => 110 => 011 => 2 = 1 + 1
[2,1,1,1]
=> 0111 => 1000 => 0001 => 4 = 3 + 1
[1,1,1,1,1]
=> 11111 => 00000 => 00000 => 6 = 5 + 1
[6]
=> 0 => 1 => 1 => 1 = 0 + 1
[5,1]
=> 11 => 00 => 00 => 3 = 2 + 1
[4,2]
=> 00 => 11 => 11 => 1 = 0 + 1
[4,1,1]
=> 011 => 100 => 001 => 3 = 2 + 1
[3,3]
=> 11 => 00 => 00 => 3 = 2 + 1
[3,2,1]
=> 101 => 010 => 001 => 3 = 2 + 1
[3,1,1,1]
=> 1111 => 0000 => 0000 => 5 = 4 + 1
[2,2,2]
=> 000 => 111 => 111 => 1 = 0 + 1
[2,2,1,1]
=> 0011 => 1100 => 0011 => 3 = 2 + 1
[2,1,1,1,1]
=> 01111 => 10000 => 00001 => 5 = 4 + 1
[1,1,1,1,1,1]
=> 111111 => 000000 => 000000 => 7 = 6 + 1
[7]
=> 1 => 0 => 0 => 2 = 1 + 1
[6,1]
=> 01 => 10 => 01 => 2 = 1 + 1
[5,2]
=> 10 => 01 => 01 => 2 = 1 + 1
[5,1,1]
=> 111 => 000 => 000 => 4 = 3 + 1
[4,3]
=> 01 => 10 => 01 => 2 = 1 + 1
[4,2,1]
=> 001 => 110 => 011 => 2 = 1 + 1
[4,1,1,1]
=> 0111 => 1000 => 0001 => 4 = 3 + 1
[3,3,1]
=> 111 => 000 => 000 => 4 = 3 + 1
[3,2,2]
=> 100 => 011 => 011 => 2 = 1 + 1
[3,2,1,1]
=> 1011 => 0100 => 0001 => 4 = 3 + 1
[3,1,1,1,1]
=> 11111 => 00000 => 00000 => 6 = 5 + 1
[2,2,2,1]
=> 0001 => 1110 => 0111 => 2 = 1 + 1
[2,2,1,1,1]
=> 00111 => 11000 => 00011 => 4 = 3 + 1
[2,1,1,1,1,1]
=> 011111 => 100000 => 000001 => 6 = 5 + 1
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => 0000000 => 8 = 7 + 1
[8]
=> 0 => 1 => 1 => 1 = 0 + 1
[7,1]
=> 11 => 00 => 00 => 3 = 2 + 1
[6,2]
=> 00 => 11 => 11 => 1 = 0 + 1
[6,1,1]
=> 011 => 100 => 001 => 3 = 2 + 1
[5,3]
=> 11 => 00 => 00 => 3 = 2 + 1
[5,2,1]
=> 101 => 010 => 001 => 3 = 2 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 00000000000 => 00000000000 => ? = 11 + 1
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 1100000000 => ? => ? ∊ {8,12} + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 000000000000 => 000000000000 => ? ∊ {8,12} + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000288
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 99%●distinct values known / distinct values provided: 85%
Values
[1]
=> 1 => 1 => 1 => 1
[2]
=> 0 => 0 => 0 => 0
[1,1]
=> 11 => 11 => 11 => 2
[3]
=> 1 => 1 => 1 => 1
[2,1]
=> 01 => 10 => 10 => 1
[1,1,1]
=> 111 => 111 => 111 => 3
[4]
=> 0 => 0 => 0 => 0
[3,1]
=> 11 => 11 => 11 => 2
[2,2]
=> 00 => 00 => 00 => 0
[2,1,1]
=> 011 => 110 => 110 => 2
[1,1,1,1]
=> 1111 => 1111 => 1111 => 4
[5]
=> 1 => 1 => 1 => 1
[4,1]
=> 01 => 10 => 10 => 1
[3,2]
=> 10 => 01 => 01 => 1
[3,1,1]
=> 111 => 111 => 111 => 3
[2,2,1]
=> 001 => 010 => 100 => 1
[2,1,1,1]
=> 0111 => 1110 => 1110 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[6]
=> 0 => 0 => 0 => 0
[5,1]
=> 11 => 11 => 11 => 2
[4,2]
=> 00 => 00 => 00 => 0
[4,1,1]
=> 011 => 110 => 110 => 2
[3,3]
=> 11 => 11 => 11 => 2
[3,2,1]
=> 101 => 011 => 011 => 2
[3,1,1,1]
=> 1111 => 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 000 => 0
[2,2,1,1]
=> 0011 => 0110 => 1010 => 2
[2,1,1,1,1]
=> 01111 => 11110 => 11110 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 6
[7]
=> 1 => 1 => 1 => 1
[6,1]
=> 01 => 10 => 10 => 1
[5,2]
=> 10 => 01 => 01 => 1
[5,1,1]
=> 111 => 111 => 111 => 3
[4,3]
=> 01 => 10 => 10 => 1
[4,2,1]
=> 001 => 010 => 100 => 1
[4,1,1,1]
=> 0111 => 1110 => 1110 => 3
[3,3,1]
=> 111 => 111 => 111 => 3
[3,2,2]
=> 100 => 001 => 001 => 1
[3,2,1,1]
=> 1011 => 0111 => 0111 => 3
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[2,2,2,1]
=> 0001 => 0010 => 1000 => 1
[2,2,1,1,1]
=> 00111 => 01110 => 10110 => 3
[2,1,1,1,1,1]
=> 011111 => 111110 => 111110 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 7
[8]
=> 0 => 0 => 0 => 0
[7,1]
=> 11 => 11 => 11 => 2
[6,2]
=> 00 => 00 => 00 => 0
[6,1,1]
=> 011 => 110 => 110 => 2
[5,3]
=> 11 => 11 => 11 => 2
[5,2,1]
=> 101 => 011 => 011 => 2
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 11111111111 => 11111111111 => ? = 11
[3,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? ∊ {10,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 111111111111 => 111111111111 => ? ∊ {10,12}
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000392
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 98%●distinct values known / distinct values provided: 85%
Mp00104: Binary words —reverse⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 98%●distinct values known / distinct values provided: 85%
Values
[1]
=> 1 => 1 => 1 => 1
[2]
=> 0 => 0 => 0 => 0
[1,1]
=> 11 => 11 => 11 => 2
[3]
=> 1 => 1 => 1 => 1
[2,1]
=> 01 => 10 => 01 => 1
[1,1,1]
=> 111 => 111 => 111 => 3
[4]
=> 0 => 0 => 0 => 0
[3,1]
=> 11 => 11 => 11 => 2
[2,2]
=> 00 => 00 => 00 => 0
[2,1,1]
=> 011 => 110 => 011 => 2
[1,1,1,1]
=> 1111 => 1111 => 1111 => 4
[5]
=> 1 => 1 => 1 => 1
[4,1]
=> 01 => 10 => 01 => 1
[3,2]
=> 10 => 01 => 01 => 1
[3,1,1]
=> 111 => 111 => 111 => 3
[2,2,1]
=> 001 => 100 => 001 => 1
[2,1,1,1]
=> 0111 => 1110 => 0111 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[6]
=> 0 => 0 => 0 => 0
[5,1]
=> 11 => 11 => 11 => 2
[4,2]
=> 00 => 00 => 00 => 0
[4,1,1]
=> 011 => 110 => 011 => 2
[3,3]
=> 11 => 11 => 11 => 2
[3,2,1]
=> 101 => 101 => 011 => 2
[3,1,1,1]
=> 1111 => 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 000 => 0
[2,2,1,1]
=> 0011 => 1100 => 0011 => 2
[2,1,1,1,1]
=> 01111 => 11110 => 01111 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 6
[7]
=> 1 => 1 => 1 => 1
[6,1]
=> 01 => 10 => 01 => 1
[5,2]
=> 10 => 01 => 01 => 1
[5,1,1]
=> 111 => 111 => 111 => 3
[4,3]
=> 01 => 10 => 01 => 1
[4,2,1]
=> 001 => 100 => 001 => 1
[4,1,1,1]
=> 0111 => 1110 => 0111 => 3
[3,3,1]
=> 111 => 111 => 111 => 3
[3,2,2]
=> 100 => 001 => 001 => 1
[3,2,1,1]
=> 1011 => 1101 => 0111 => 3
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[2,2,2,1]
=> 0001 => 1000 => 0001 => 1
[2,2,1,1,1]
=> 00111 => 11100 => 00111 => 3
[2,1,1,1,1,1]
=> 011111 => 111110 => 011111 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 7
[8]
=> 0 => 0 => 0 => 0
[7,1]
=> 11 => 11 => 11 => 2
[6,2]
=> 00 => 00 => 00 => 0
[6,1,1]
=> 011 => 110 => 011 => 2
[5,3]
=> 11 => 11 => 11 => 2
[5,2,1]
=> 101 => 101 => 011 => 2
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 11111111111 => ? => ? = 11
[3,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? ∊ {8,10,12}
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 1111111100 => ? => ? ∊ {8,10,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 111111111111 => 111111111111 => ? ∊ {8,10,12}
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001372
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 98%●distinct values known / distinct values provided: 85%
Mp00104: Binary words —reverse⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 98%●distinct values known / distinct values provided: 85%
Values
[1]
=> 1 => 1 => 1 => 1
[2]
=> 0 => 0 => 0 => 0
[1,1]
=> 11 => 11 => 11 => 2
[3]
=> 1 => 1 => 1 => 1
[2,1]
=> 01 => 10 => 01 => 1
[1,1,1]
=> 111 => 111 => 111 => 3
[4]
=> 0 => 0 => 0 => 0
[3,1]
=> 11 => 11 => 11 => 2
[2,2]
=> 00 => 00 => 00 => 0
[2,1,1]
=> 011 => 110 => 011 => 2
[1,1,1,1]
=> 1111 => 1111 => 1111 => 4
[5]
=> 1 => 1 => 1 => 1
[4,1]
=> 01 => 10 => 01 => 1
[3,2]
=> 10 => 01 => 01 => 1
[3,1,1]
=> 111 => 111 => 111 => 3
[2,2,1]
=> 001 => 100 => 001 => 1
[2,1,1,1]
=> 0111 => 1110 => 0111 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[6]
=> 0 => 0 => 0 => 0
[5,1]
=> 11 => 11 => 11 => 2
[4,2]
=> 00 => 00 => 00 => 0
[4,1,1]
=> 011 => 110 => 011 => 2
[3,3]
=> 11 => 11 => 11 => 2
[3,2,1]
=> 101 => 101 => 011 => 2
[3,1,1,1]
=> 1111 => 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 000 => 0
[2,2,1,1]
=> 0011 => 1100 => 0011 => 2
[2,1,1,1,1]
=> 01111 => 11110 => 01111 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 6
[7]
=> 1 => 1 => 1 => 1
[6,1]
=> 01 => 10 => 01 => 1
[5,2]
=> 10 => 01 => 01 => 1
[5,1,1]
=> 111 => 111 => 111 => 3
[4,3]
=> 01 => 10 => 01 => 1
[4,2,1]
=> 001 => 100 => 001 => 1
[4,1,1,1]
=> 0111 => 1110 => 0111 => 3
[3,3,1]
=> 111 => 111 => 111 => 3
[3,2,2]
=> 100 => 001 => 001 => 1
[3,2,1,1]
=> 1011 => 1101 => 0111 => 3
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[2,2,2,1]
=> 0001 => 1000 => 0001 => 1
[2,2,1,1,1]
=> 00111 => 11100 => 00111 => 3
[2,1,1,1,1,1]
=> 011111 => 111110 => 011111 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 7
[8]
=> 0 => 0 => 0 => 0
[7,1]
=> 11 => 11 => 11 => 2
[6,2]
=> 00 => 00 => 00 => 0
[6,1,1]
=> 011 => 110 => 011 => 2
[5,3]
=> 11 => 11 => 11 => 2
[5,2,1]
=> 101 => 101 => 011 => 2
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? = 10
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 11111111111 => ? => ? = 11
[3,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? ∊ {8,10,12}
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 1111111100 => ? => ? ∊ {8,10,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 111111111111 => 111111111111 => ? ∊ {8,10,12}
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: St001419
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 97%●distinct values known / distinct values provided: 77%
Mp00104: Binary words —reverse⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St001419: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 97%●distinct values known / distinct values provided: 77%
Values
[1]
=> 1 => 1 => 1 => 1
[2]
=> 0 => 0 => 0 => 0
[1,1]
=> 11 => 11 => 11 => 2
[3]
=> 1 => 1 => 1 => 1
[2,1]
=> 01 => 10 => 01 => 1
[1,1,1]
=> 111 => 111 => 111 => 3
[4]
=> 0 => 0 => 0 => 0
[3,1]
=> 11 => 11 => 11 => 2
[2,2]
=> 00 => 00 => 00 => 0
[2,1,1]
=> 011 => 110 => 011 => 2
[1,1,1,1]
=> 1111 => 1111 => 1111 => 4
[5]
=> 1 => 1 => 1 => 1
[4,1]
=> 01 => 10 => 01 => 1
[3,2]
=> 10 => 01 => 01 => 1
[3,1,1]
=> 111 => 111 => 111 => 3
[2,2,1]
=> 001 => 100 => 001 => 1
[2,1,1,1]
=> 0111 => 1110 => 0111 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[6]
=> 0 => 0 => 0 => 0
[5,1]
=> 11 => 11 => 11 => 2
[4,2]
=> 00 => 00 => 00 => 0
[4,1,1]
=> 011 => 110 => 011 => 2
[3,3]
=> 11 => 11 => 11 => 2
[3,2,1]
=> 101 => 101 => 011 => 2
[3,1,1,1]
=> 1111 => 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 000 => 0
[2,2,1,1]
=> 0011 => 1100 => 0011 => 2
[2,1,1,1,1]
=> 01111 => 11110 => 01111 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 6
[7]
=> 1 => 1 => 1 => 1
[6,1]
=> 01 => 10 => 01 => 1
[5,2]
=> 10 => 01 => 01 => 1
[5,1,1]
=> 111 => 111 => 111 => 3
[4,3]
=> 01 => 10 => 01 => 1
[4,2,1]
=> 001 => 100 => 001 => 1
[4,1,1,1]
=> 0111 => 1110 => 0111 => 3
[3,3,1]
=> 111 => 111 => 111 => 3
[3,2,2]
=> 100 => 001 => 001 => 1
[3,2,1,1]
=> 1011 => 1101 => 0111 => 3
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[2,2,2,1]
=> 0001 => 1000 => 0001 => 1
[2,2,1,1,1]
=> 00111 => 11100 => 00111 => 3
[2,1,1,1,1,1]
=> 011111 => 111110 => 011111 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 7
[8]
=> 0 => 0 => 0 => 0
[7,1]
=> 11 => 11 => 11 => 2
[6,2]
=> 00 => 00 => 00 => 0
[6,1,1]
=> 011 => 110 => 011 => 2
[5,3]
=> 11 => 11 => 11 => 2
[5,2,1]
=> 101 => 101 => 011 => 2
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? = 10
[2,1,1,1,1,1,1,1,1,1]
=> 0111111111 => 1111111110 => 0111111111 => ? ∊ {9,11}
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => 11111111111 => ? => ? ∊ {9,11}
[3,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => 1111111111 => ? ∊ {8,10,10,12}
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => 1111111100 => ? => ? ∊ {8,10,10,12}
[2,1,1,1,1,1,1,1,1,1,1]
=> 01111111111 => 11111111110 => 01111111111 => ? ∊ {8,10,10,12}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 111111111111 => 111111111111 => ? ∊ {8,10,10,12}
Description
The length of the longest palindromic factor beginning with a one of a binary word.
Matching statistic: St000097
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: 77% ●values known / values provided: 94%●distinct values known / distinct values provided: 77%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 77% ●values known / values provided: 94%●distinct values known / distinct values provided: 77%
Values
[1]
=> 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2]
=> 0 => [2] => ([],2)
=> 1 = 0 + 1
[1,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3]
=> 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[2,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4]
=> 0 => [2] => ([],2)
=> 1 = 0 + 1
[3,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[2,2]
=> 00 => [3] => ([],3)
=> 1 = 0 + 1
[2,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[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 = 4 + 1
[5]
=> 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[4,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[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 = 3 + 1
[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 = 5 + 1
[6]
=> 0 => [2] => ([],2)
=> 1 = 0 + 1
[5,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[4,2]
=> 00 => [3] => ([],3)
=> 1 = 0 + 1
[4,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[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 = 4 + 1
[2,2,2]
=> 000 => [4] => ([],4)
=> 1 = 0 + 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
[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 = 4 + 1
[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 = 6 + 1
[7]
=> 1 => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
[6,1]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[5,1,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[4,3]
=> 01 => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,1,1,1]
=> 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 = 3 + 1
[3,3,1]
=> 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> 2 = 1 + 1
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[3,1,1,1,1]
=> 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 = 5 + 1
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,2,1,1,1]
=> 00111 => [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)
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> 011111 => [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)
=> 6 = 5 + 1
[1,1,1,1,1,1,1]
=> 1111111 => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 8 = 7 + 1
[8]
=> 0 => [2] => ([],2)
=> 1 = 0 + 1
[7,1]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[6,2]
=> 00 => [3] => ([],3)
=> 1 = 0 + 1
[6,1,1]
=> 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5,3]
=> 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,2,1,1,1,1,1,1]
=> 00111111 => [3,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {6,10} + 1
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => [1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? ∊ {6,10} + 1
[3,2,1,1,1,1,1,1]
=> 10111111 => [1,2,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {5,7,7,9,11} + 1
[2,2,2,1,1,1,1,1]
=> 00011111 => [4,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {5,7,7,9,11} + 1
[2,2,1,1,1,1,1,1,1]
=> 001111111 => [3,1,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {5,7,7,9,11} + 1
[2,1,1,1,1,1,1,1,1,1]
=> 0111111111 => [2,1,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? ∊ {5,7,7,9,11} + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => [1,1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {5,7,7,9,11} + 1
[4,2,1,1,1,1,1,1]
=> 00111111 => [3,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {4,6,6,6,8,8,10,10,12} + 1
[3,2,2,1,1,1,1,1]
=> 10011111 => [1,3,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {4,6,6,6,8,8,10,10,12} + 1
[3,2,1,1,1,1,1,1,1]
=> 101111111 => [1,2,1,1,1,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {4,6,6,6,8,8,10,10,12} + 1
[3,1,1,1,1,1,1,1,1,1]
=> 1111111111 => [1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? ∊ {4,6,6,6,8,8,10,10,12} + 1
[2,2,2,2,1,1,1,1]
=> 00001111 => [5,1,1,1,1] => ([(0,5),(0,6),(0,7),(0,8),(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {4,6,6,6,8,8,10,10,12} + 1
[2,2,2,1,1,1,1,1,1]
=> 000111111 => [4,1,1,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {4,6,6,6,8,8,10,10,12} + 1
[2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => [3,1,1,1,1,1,1,1,1] => ?
=> ? ∊ {4,6,6,6,8,8,10,10,12} + 1
[2,1,1,1,1,1,1,1,1,1,1]
=> 01111111111 => [2,1,1,1,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? ∊ {4,6,6,6,8,8,10,10,12} + 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => [1,1,1,1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(2,10),(2,11),(2,12),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? ∊ {4,6,6,6,8,8,10,10,12} + 1
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.
The following 28 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000098The chromatic number of a graph. St001581The achromatic number of a graph. St001494The Alon-Tarsi number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. 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. St000822The Hadwiger number of the graph. St001812The biclique partition number of a graph. St000475The number of parts equal to 1 in a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St000022The number of fixed points of a permutation. St001330The hat guessing number of a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St000696The number of cycles in the breakpoint graph of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000247The number of singleton blocks of a set partition. St000241The number of cyclical small excedances. St000894The trace of an alternating sign matrix.
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