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Your data matches 39 different statistics following compositions of up to 3 maps.
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Matching statistic: St000148
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(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: St000022
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
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%
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] => 2
[1,1]
=> [[1],[2]]
=> [2,1] => 0
[3]
=> [[1,2,3]]
=> [1,2,3] => 3
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 4
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 5
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 3
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 4
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => 0
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 7
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => 5
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => 3
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => 5
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => 3
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => 3
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 8
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => 6
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => 4
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => 6
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => 2
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => 4
Description
The number of fixed points of a permutation.
Matching statistic: St000475
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1]
=> 1
[2]
=> [[1,2]]
=> [1,2] => [1,1]
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => [2]
=> 0
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,1,1]
=> 3
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1]
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,1]
=> 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,1]
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,2]
=> 0
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [2,1,1]
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [2,2]
=> 0
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,2,1]
=> 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,1,1,1]
=> 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,2,1]
=> 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [2,2,1]
=> 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [2,2,1]
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,1,1,1]
=> 4
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [2,2,1,1]
=> 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,1,1,1,1]
=> 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [2,2,2]
=> 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,2,1,1]
=> 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [2,2,1,1]
=> 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,2,1,1]
=> 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [2,2,2]
=> 0
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [2,2,1,1]
=> 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [2,2,2]
=> 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> 7
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,1,1,1,1]
=> 5
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [2,2,1,1,1]
=> 3
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [2,1,1,1,1,1]
=> 5
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [2,2,2,1]
=> 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,2,1,1,1]
=> 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [2,2,1,1,1]
=> 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,2,2,1]
=> 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [2,2,1,1,1]
=> 3
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [2,2,2,1]
=> 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [2,2,1,1,1]
=> 3
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [2,2,2,1]
=> 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [2,2,2,1]
=> 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [2,2,2,1]
=> 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [2,2,2,1]
=> 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,1,1,1,1,1]
=> 6
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [2,2,1,1,1,1]
=> 4
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [2,1,1,1,1,1,1]
=> 6
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [2,2,2,1,1]
=> 2
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [2,2,1,1,1,1]
=> 4
Description
The number of parts equal to 1 in a 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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: St001247
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1]
=> 1
[2]
=> [[1,2]]
=> [1,2] => [1,1]
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => [2]
=> 0
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,1,1]
=> 3
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1]
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,1]
=> 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,1]
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,2]
=> 0
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [2,1,1]
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [2,2]
=> 0
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,2,1]
=> 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,1,1,1]
=> 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,2,1]
=> 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [2,2,1]
=> 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [2,2,1]
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,1,1,1]
=> 4
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [2,2,1,1]
=> 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,1,1,1,1]
=> 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [2,2,2]
=> 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,2,1,1]
=> 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [2,2,1,1]
=> 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,2,1,1]
=> 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [2,2,2]
=> 0
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [2,2,1,1]
=> 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [2,2,2]
=> 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> 7
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,1,1,1,1]
=> 5
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [2,2,1,1,1]
=> 3
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [2,1,1,1,1,1]
=> 5
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [2,2,2,1]
=> 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,2,1,1,1]
=> 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [2,2,1,1,1]
=> 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,2,2,1]
=> 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [2,2,1,1,1]
=> 3
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [2,2,2,1]
=> 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [2,2,1,1,1]
=> 3
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [2,2,2,1]
=> 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [2,2,2,1]
=> 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [2,2,2,1]
=> 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [2,2,2,1]
=> 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,1,1,1,1,1]
=> 6
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [2,2,1,1,1,1]
=> 4
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [2,1,1,1,1,1,1]
=> 6
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [2,2,2,1,1]
=> 2
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [2,2,1,1,1,1]
=> 4
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St001249
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1]
=> 1
[2]
=> [[1,2]]
=> [1,2] => [1,1]
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => [2]
=> 0
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,1,1]
=> 3
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1]
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,1]
=> 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> 4
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,1]
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,2]
=> 0
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [2,1,1]
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [2,2]
=> 0
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 5
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,1,1]
=> 3
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,2,1]
=> 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,1,1,1]
=> 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,2,1]
=> 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [2,2,1]
=> 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [2,2,1]
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,1,1,1]
=> 4
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [2,2,1,1]
=> 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,1,1,1,1]
=> 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [2,2,2]
=> 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,2,1,1]
=> 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [2,2,1,1]
=> 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,2,1,1]
=> 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [2,2,2]
=> 0
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [2,2,1,1]
=> 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [2,2,2]
=> 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> 7
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,1,1,1,1]
=> 5
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [2,2,1,1,1]
=> 3
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [2,1,1,1,1,1]
=> 5
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [2,2,2,1]
=> 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,2,1,1,1]
=> 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [2,2,1,1,1]
=> 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,2,2,1]
=> 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [2,2,1,1,1]
=> 3
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [2,2,2,1]
=> 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [2,2,1,1,1]
=> 3
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [2,2,2,1]
=> 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [2,2,2,1]
=> 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [2,2,2,1]
=> 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [2,2,2,1]
=> 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,1,1,1,1,1]
=> 6
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [2,2,1,1,1,1]
=> 4
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [2,1,1,1,1,1,1]
=> 6
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [2,2,2,1,1]
=> 2
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [2,2,1,1,1,1]
=> 4
Description
Sum of the odd parts of a partition.
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
Description
The length of the partition.
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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
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
St000288: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
St000288: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 99%●distinct values known / distinct values provided: 91%
Values
[1]
=> 1 => 1
[2]
=> 0 => 0
[1,1]
=> 11 => 2
[3]
=> 1 => 1
[2,1]
=> 01 => 1
[1,1,1]
=> 111 => 3
[4]
=> 0 => 0
[3,1]
=> 11 => 2
[2,2]
=> 00 => 0
[2,1,1]
=> 011 => 2
[1,1,1,1]
=> 1111 => 4
[5]
=> 1 => 1
[4,1]
=> 01 => 1
[3,2]
=> 10 => 1
[3,1,1]
=> 111 => 3
[2,2,1]
=> 001 => 1
[2,1,1,1]
=> 0111 => 3
[1,1,1,1,1]
=> 11111 => 5
[6]
=> 0 => 0
[5,1]
=> 11 => 2
[4,2]
=> 00 => 0
[4,1,1]
=> 011 => 2
[3,3]
=> 11 => 2
[3,2,1]
=> 101 => 2
[3,1,1,1]
=> 1111 => 4
[2,2,2]
=> 000 => 0
[2,2,1,1]
=> 0011 => 2
[2,1,1,1,1]
=> 01111 => 4
[1,1,1,1,1,1]
=> 111111 => 6
[7]
=> 1 => 1
[6,1]
=> 01 => 1
[5,2]
=> 10 => 1
[5,1,1]
=> 111 => 3
[4,3]
=> 01 => 1
[4,2,1]
=> 001 => 1
[4,1,1,1]
=> 0111 => 3
[3,3,1]
=> 111 => 3
[3,2,2]
=> 100 => 1
[3,2,1,1]
=> 1011 => 3
[3,1,1,1,1]
=> 11111 => 5
[2,2,2,1]
=> 0001 => 1
[2,2,1,1,1]
=> 00111 => 3
[2,1,1,1,1,1]
=> 011111 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 7
[8]
=> 0 => 0
[7,1]
=> 11 => 2
[6,2]
=> 00 => 0
[6,1,1]
=> 011 => 2
[5,3]
=> 11 => 2
[5,2,1]
=> 101 => 2
[1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => ? = 10
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 29 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000097The order of the largest clique of the graph. St000098The chromatic 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. 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. St000806The semiperimeter of the associated bargraph. 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. St000696The number of cycles in the breakpoint graph of a permutation. St001812The biclique partition number of a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. 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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