Your data matches 43 different statistics following compositions of up to 3 maps.
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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 $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Mp00317: Integer partitions odd partsBinary words
St000288: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
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
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Mp00317: Integer partitions odd partsBinary words
St001372: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
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
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.
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
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.
Mp00317: Integer partitions odd partsBinary words
Mp00224: Binary words runsortBinary 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]
=> 0 => 0 => 0
[1,1]
=> 11 => 11 => 2
[3]
=> 1 => 1 => 1
[2,1]
=> 01 => 01 => 1
[1,1,1]
=> 111 => 111 => 3
[4]
=> 0 => 0 => 0
[3,1]
=> 11 => 11 => 2
[2,2]
=> 00 => 00 => 0
[2,1,1]
=> 011 => 011 => 2
[1,1,1,1]
=> 1111 => 1111 => 4
[5]
=> 1 => 1 => 1
[4,1]
=> 01 => 01 => 1
[3,2]
=> 10 => 01 => 1
[3,1,1]
=> 111 => 111 => 3
[2,2,1]
=> 001 => 001 => 1
[2,1,1,1]
=> 0111 => 0111 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 5
[6]
=> 0 => 0 => 0
[5,1]
=> 11 => 11 => 2
[4,2]
=> 00 => 00 => 0
[4,1,1]
=> 011 => 011 => 2
[3,3]
=> 11 => 11 => 2
[3,2,1]
=> 101 => 011 => 2
[3,1,1,1]
=> 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 0
[2,2,1,1]
=> 0011 => 0011 => 2
[2,1,1,1,1]
=> 01111 => 01111 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 6
[7]
=> 1 => 1 => 1
[6,1]
=> 01 => 01 => 1
[5,2]
=> 10 => 01 => 1
[5,1,1]
=> 111 => 111 => 3
[4,3]
=> 01 => 01 => 1
[4,2,1]
=> 001 => 001 => 1
[4,1,1,1]
=> 0111 => 0111 => 3
[3,3,1]
=> 111 => 111 => 3
[3,2,2]
=> 100 => 001 => 1
[3,2,1,1]
=> 1011 => 0111 => 3
[3,1,1,1,1]
=> 11111 => 11111 => 5
[2,2,2,1]
=> 0001 => 0001 => 1
[2,2,1,1,1]
=> 00111 => 00111 => 3
[2,1,1,1,1,1]
=> 011111 => 011111 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 7
[8]
=> 0 => 0 => 0
[7,1]
=> 11 => 11 => 2
[6,2]
=> 00 => 00 => 0
[6,1,1]
=> 011 => 011 => 2
[5,3]
=> 11 => 11 => 2
[5,2,1]
=> 101 => 011 => 2
Description
The length of the longest run of ones in a binary word.
Mp00317: Integer partitions odd partsBinary words
Mp00224: Binary words runsortBinary 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]
=> 0 => 0 => 0
[1,1]
=> 11 => 11 => 2
[3]
=> 1 => 1 => 1
[2,1]
=> 01 => 01 => 1
[1,1,1]
=> 111 => 111 => 3
[4]
=> 0 => 0 => 0
[3,1]
=> 11 => 11 => 2
[2,2]
=> 00 => 00 => 0
[2,1,1]
=> 011 => 011 => 2
[1,1,1,1]
=> 1111 => 1111 => 4
[5]
=> 1 => 1 => 1
[4,1]
=> 01 => 01 => 1
[3,2]
=> 10 => 01 => 1
[3,1,1]
=> 111 => 111 => 3
[2,2,1]
=> 001 => 001 => 1
[2,1,1,1]
=> 0111 => 0111 => 3
[1,1,1,1,1]
=> 11111 => 11111 => 5
[6]
=> 0 => 0 => 0
[5,1]
=> 11 => 11 => 2
[4,2]
=> 00 => 00 => 0
[4,1,1]
=> 011 => 011 => 2
[3,3]
=> 11 => 11 => 2
[3,2,1]
=> 101 => 011 => 2
[3,1,1,1]
=> 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 0
[2,2,1,1]
=> 0011 => 0011 => 2
[2,1,1,1,1]
=> 01111 => 01111 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 6
[7]
=> 1 => 1 => 1
[6,1]
=> 01 => 01 => 1
[5,2]
=> 10 => 01 => 1
[5,1,1]
=> 111 => 111 => 3
[4,3]
=> 01 => 01 => 1
[4,2,1]
=> 001 => 001 => 1
[4,1,1,1]
=> 0111 => 0111 => 3
[3,3,1]
=> 111 => 111 => 3
[3,2,2]
=> 100 => 001 => 1
[3,2,1,1]
=> 1011 => 0111 => 3
[3,1,1,1,1]
=> 11111 => 11111 => 5
[2,2,2,1]
=> 0001 => 0001 => 1
[2,2,1,1,1]
=> 00111 => 00111 => 3
[2,1,1,1,1,1]
=> 011111 => 011111 => 5
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 7
[8]
=> 0 => 0 => 0
[7,1]
=> 11 => 11 => 2
[6,2]
=> 00 => 00 => 0
[6,1,1]
=> 011 => 011 => 2
[5,3]
=> 11 => 11 => 2
[5,2,1]
=> 101 => 011 => 2
Description
The length of the longest palindromic factor beginning with a one of a binary word.
Matching statistic: St000297
Mp00317: Integer partitions odd partsBinary words
Mp00224: Binary words runsortBinary words
Mp00104: Binary words reverseBinary words
St000297: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
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 => 01 => 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 => 011 => 110 => 2
[1,1,1,1]
=> 1111 => 1111 => 1111 => 4
[5]
=> 1 => 1 => 1 => 1
[4,1]
=> 01 => 01 => 10 => 1
[3,2]
=> 10 => 01 => 10 => 1
[3,1,1]
=> 111 => 111 => 111 => 3
[2,2,1]
=> 001 => 001 => 100 => 1
[2,1,1,1]
=> 0111 => 0111 => 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 => 011 => 110 => 2
[3,3]
=> 11 => 11 => 11 => 2
[3,2,1]
=> 101 => 011 => 110 => 2
[3,1,1,1]
=> 1111 => 1111 => 1111 => 4
[2,2,2]
=> 000 => 000 => 000 => 0
[2,2,1,1]
=> 0011 => 0011 => 1100 => 2
[2,1,1,1,1]
=> 01111 => 01111 => 11110 => 4
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 6
[7]
=> 1 => 1 => 1 => 1
[6,1]
=> 01 => 01 => 10 => 1
[5,2]
=> 10 => 01 => 10 => 1
[5,1,1]
=> 111 => 111 => 111 => 3
[4,3]
=> 01 => 01 => 10 => 1
[4,2,1]
=> 001 => 001 => 100 => 1
[4,1,1,1]
=> 0111 => 0111 => 1110 => 3
[3,3,1]
=> 111 => 111 => 111 => 3
[3,2,2]
=> 100 => 001 => 100 => 1
[3,2,1,1]
=> 1011 => 0111 => 1110 => 3
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 5
[2,2,2,1]
=> 0001 => 0001 => 1000 => 1
[2,2,1,1,1]
=> 00111 => 00111 => 11100 => 3
[2,1,1,1,1,1]
=> 011111 => 011111 => 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 => 011 => 110 => 2
[5,3]
=> 11 => 11 => 11 => 2
[5,2,1]
=> 101 => 011 => 110 => 2
Description
The number of leading ones in a binary word.
Matching statistic: St000475
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00108: Permutations cycle typeInteger 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 partsBinary words
Mp00105: Binary words complementBinary words
Mp00224: Binary words runsortBinary 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].
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St000010The length of the partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. 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. 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. St000806The semiperimeter of the associated bargraph. St000822The Hadwiger number of the graph. St000696The number of cycles in the breakpoint graph of a permutation. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001812The biclique partition number of a graph. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001903The number of fixed points of a parking function. St000884The number of isolated descents of a permutation.