Your data matches 24 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000992
(load all 2 compositions to match this statistic)
Mapping
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[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
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: St000148
(load all 3 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => [1,1]
 => 2
[1,1]
 => [2]
 => 0
[3]
 => [1,1,1]
 => 3
[2,1]
 => [2,1]
 => 1
[1,1,1]
 => [3]
 => 1
[4]
 => [1,1,1,1]
 => 4
[3,1]
 => [2,1,1]
 => 2
[2,2]
 => [2,2]
 => 0
[2,1,1]
 => [3,1]
 => 2
[1,1,1,1]
 => [4]
 => 0
[5]
 => [1,1,1,1,1]
 => 5
[4,1]
 => [2,1,1,1]
 => 3
[3,2]
 => [2,2,1]
 => 1
[3,1,1]
 => [3,1,1]
 => 3
[2,2,1]
 => [3,2]
 => 1
[2,1,1,1]
 => [4,1]
 => 1
[1,1,1,1,1]
 => [5]
 => 1
[6]
 => [1,1,1,1,1,1]
 => 6
[5,1]
 => [2,1,1,1,1]
 => 4
[4,2]
 => [2,2,1,1]
 => 2
[4,1,1]
 => [3,1,1,1]
 => 4
[3,3]
 => [2,2,2]
 => 0
[3,2,1]
 => [3,2,1]
 => 2
[3,1,1,1]
 => [4,1,1]
 => 2
[2,2,2]
 => [3,3]
 => 2
[2,2,1,1]
 => [4,2]
 => 0
[2,1,1,1,1]
 => [5,1]
 => 2
[1,1,1,1,1,1]
 => [6]
 => 0
[7]
 => [1,1,1,1,1,1,1]
 => 7
[6,1]
 => [2,1,1,1,1,1]
 => 5
[5,2]
 => [2,2,1,1,1]
 => 3
[5,1,1]
 => [3,1,1,1,1]
 => 5
[4,3]
 => [2,2,2,1]
 => 1
[4,2,1]
 => [3,2,1,1]
 => 3
[4,1,1,1]
 => [4,1,1,1]
 => 3
[3,3,1]
 => [3,2,2]
 => 1
[3,2,2]
 => [3,3,1]
 => 3
[3,2,1,1]
 => [4,2,1]
 => 1
[3,1,1,1,1]
 => [5,1,1]
 => 3
[2,2,2,1]
 => [4,3]
 => 1
[2,2,1,1,1]
 => [5,2]
 => 1
[2,1,1,1,1,1]
 => [6,1]
 => 1
[1,1,1,1,1,1,1]
 => [7]
 => 1
Description
The number of odd parts of a partition.
Matching statistic: St000022
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
[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
Description
The number of fixed points of a permutation.
Matching statistic: St000288
(load all 8 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
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
[2]
 => [1,1]
 => 11 => 2
[1,1]
 => [2]
 => 0 => 0
[3]
 => [1,1,1]
 => 111 => 3
[2,1]
 => [2,1]
 => 01 => 1
[1,1,1]
 => [3]
 => 1 => 1
[4]
 => [1,1,1,1]
 => 1111 => 4
[3,1]
 => [2,1,1]
 => 011 => 2
[2,2]
 => [2,2]
 => 00 => 0
[2,1,1]
 => [3,1]
 => 11 => 2
[1,1,1,1]
 => [4]
 => 0 => 0
[5]
 => [1,1,1,1,1]
 => 11111 => 5
[4,1]
 => [2,1,1,1]
 => 0111 => 3
[3,2]
 => [2,2,1]
 => 001 => 1
[3,1,1]
 => [3,1,1]
 => 111 => 3
[2,2,1]
 => [3,2]
 => 10 => 1
[2,1,1,1]
 => [4,1]
 => 01 => 1
[1,1,1,1,1]
 => [5]
 => 1 => 1
[6]
 => [1,1,1,1,1,1]
 => 111111 => 6
[5,1]
 => [2,1,1,1,1]
 => 01111 => 4
[4,2]
 => [2,2,1,1]
 => 0011 => 2
[4,1,1]
 => [3,1,1,1]
 => 1111 => 4
[3,3]
 => [2,2,2]
 => 000 => 0
[3,2,1]
 => [3,2,1]
 => 101 => 2
[3,1,1,1]
 => [4,1,1]
 => 011 => 2
[2,2,2]
 => [3,3]
 => 11 => 2
[2,2,1,1]
 => [4,2]
 => 00 => 0
[2,1,1,1,1]
 => [5,1]
 => 11 => 2
[1,1,1,1,1,1]
 => [6]
 => 0 => 0
[7]
 => [1,1,1,1,1,1,1]
 => 1111111 => 7
[6,1]
 => [2,1,1,1,1,1]
 => 011111 => 5
[5,2]
 => [2,2,1,1,1]
 => 00111 => 3
[5,1,1]
 => [3,1,1,1,1]
 => 11111 => 5
[4,3]
 => [2,2,2,1]
 => 0001 => 1
[4,2,1]
 => [3,2,1,1]
 => 1011 => 3
[4,1,1,1]
 => [4,1,1,1]
 => 0111 => 3
[3,3,1]
 => [3,2,2]
 => 100 => 1
[3,2,2]
 => [3,3,1]
 => 111 => 3
[3,2,1,1]
 => [4,2,1]
 => 001 => 1
[3,1,1,1,1]
 => [5,1,1]
 => 111 => 3
[2,2,2,1]
 => [4,3]
 => 01 => 1
[2,2,1,1,1]
 => [5,2]
 => 10 => 1
[2,1,1,1,1,1]
 => [6,1]
 => 01 => 1
[1,1,1,1,1,1,1]
 => [7]
 => 1 => 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 7 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
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
[2]
 => [1,1]
 => 11 => 2
[1,1]
 => [2]
 => 0 => 0
[3]
 => [1,1,1]
 => 111 => 3
[2,1]
 => [2,1]
 => 01 => 1
[1,1,1]
 => [3]
 => 1 => 1
[4]
 => [1,1,1,1]
 => 1111 => 4
[3,1]
 => [2,1,1]
 => 011 => 2
[2,2]
 => [2,2]
 => 00 => 0
[2,1,1]
 => [3,1]
 => 11 => 2
[1,1,1,1]
 => [4]
 => 0 => 0
[5]
 => [1,1,1,1,1]
 => 11111 => 5
[4,1]
 => [2,1,1,1]
 => 0111 => 3
[3,2]
 => [2,2,1]
 => 001 => 1
[3,1,1]
 => [3,1,1]
 => 111 => 3
[2,2,1]
 => [3,2]
 => 10 => 1
[2,1,1,1]
 => [4,1]
 => 01 => 1
[1,1,1,1,1]
 => [5]
 => 1 => 1
[6]
 => [1,1,1,1,1,1]
 => 111111 => 6
[5,1]
 => [2,1,1,1,1]
 => 01111 => 4
[4,2]
 => [2,2,1,1]
 => 0011 => 2
[4,1,1]
 => [3,1,1,1]
 => 1111 => 4
[3,3]
 => [2,2,2]
 => 000 => 0
[3,2,1]
 => [3,2,1]
 => 101 => 2
[3,1,1,1]
 => [4,1,1]
 => 011 => 2
[2,2,2]
 => [3,3]
 => 11 => 2
[2,2,1,1]
 => [4,2]
 => 00 => 0
[2,1,1,1,1]
 => [5,1]
 => 11 => 2
[1,1,1,1,1,1]
 => [6]
 => 0 => 0
[7]
 => [1,1,1,1,1,1,1]
 => 1111111 => 7
[6,1]
 => [2,1,1,1,1,1]
 => 011111 => 5
[5,2]
 => [2,2,1,1,1]
 => 00111 => 3
[5,1,1]
 => [3,1,1,1,1]
 => 11111 => 5
[4,3]
 => [2,2,2,1]
 => 0001 => 1
[4,2,1]
 => [3,2,1,1]
 => 1011 => 3
[4,1,1,1]
 => [4,1,1,1]
 => 0111 => 3
[3,3,1]
 => [3,2,2]
 => 100 => 1
[3,2,2]
 => [3,3,1]
 => 111 => 3
[3,2,1,1]
 => [4,2,1]
 => 001 => 1
[3,1,1,1,1]
 => [5,1,1]
 => 111 => 3
[2,2,2,1]
 => [4,3]
 => 01 => 1
[2,2,1,1,1]
 => [5,2]
 => 10 => 1
[2,1,1,1,1,1]
 => [6,1]
 => 01 => 1
[1,1,1,1,1,1,1]
 => [7]
 => 1 => 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: St000247
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
 => [[1,2]]
 => [1,2] => {{1},{2}}
 => 2
[1,1]
 => [[1],[2]]
 => [2,1] => {{1,2}}
 => 0
[3]
 => [[1,2,3]]
 => [1,2,3] => {{1},{2},{3}}
 => 3
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => {{1,2},{3}}
 => 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => {{1,3},{2}}
 => 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => {{1},{2},{3},{4}}
 => 4
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => {{1,2},{3},{4}}
 => 2
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => {{1,3},{2,4}}
 => 0
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => {{1,3},{2},{4}}
 => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => {{1,4},{2,3}}
 => 0
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
 => 5
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
 => 3
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => {{1,3},{2,4},{5}}
 => 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
 => 3
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => {{1,4},{2},{3,5}}
 => 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => {{1,4},{2,3},{5}}
 => 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => {{1,5},{2,4},{3}}
 => 1
[6]
 => [[1,2,3,4,5,6]]
 => [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] => {{1,2},{3},{4},{5},{6}}
 => 4
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => {{1,3},{2,4},{5},{6}}
 => 2
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => {{1,3},{2},{4},{5},{6}}
 => 4
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => {{1,4},{2,5},{3,6}}
 => 0
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => {{1,4},{2},{3,5},{6}}
 => 2
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => {{1,4},{2,3},{5},{6}}
 => 2
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => {{1,5},{2,6},{3},{4}}
 => 2
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => {{1,5},{2,3},{4,6}}
 => 0
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => {{1,5},{2,4},{3},{6}}
 => 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => {{1,6},{2,5},{3,4}}
 => 0
[7]
 => [[1,2,3,4,5,6,7]]
 => [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] => {{1,2},{3},{4},{5},{6},{7}}
 => 5
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => {{1,3},{2,4},{5},{6},{7}}
 => 3
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => {{1,3},{2},{4},{5},{6},{7}}
 => 5
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => {{1,4},{2,5},{3,6},{7}}
 => 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => {{1,4},{2},{3,5},{6},{7}}
 => 3
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => {{1,4},{2,3},{5},{6},{7}}
 => 3
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => {{1,5},{2},{3,6},{4,7}}
 => 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => {{1,5},{2,6},{3},{4},{7}}
 => 3
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7] => {{1,5},{2,3},{4,6},{7}}
 => 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => {{1,5},{2,4},{3},{6},{7}}
 => 3
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => {{1,6},{2,4},{3,7},{5}}
 => 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => {{1,6},{2,4},{3},{5,7}}
 => 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => {{1,6},{2,5},{3,4},{7}}
 => 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => {{1,7},{2,6},{3,5},{4}}
 => 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000392
(load all 2 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
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
[2]
 => [1,1]
 => 11 => 11 => 2
[1,1]
 => [2]
 => 0 => 0 => 0
[3]
 => [1,1,1]
 => 111 => 111 => 3
[2,1]
 => [2,1]
 => 01 => 01 => 1
[1,1,1]
 => [3]
 => 1 => 1 => 1
[4]
 => [1,1,1,1]
 => 1111 => 1111 => 4
[3,1]
 => [2,1,1]
 => 011 => 011 => 2
[2,2]
 => [2,2]
 => 00 => 00 => 0
[2,1,1]
 => [3,1]
 => 11 => 11 => 2
[1,1,1,1]
 => [4]
 => 0 => 0 => 0
[5]
 => [1,1,1,1,1]
 => 11111 => 11111 => 5
[4,1]
 => [2,1,1,1]
 => 0111 => 0111 => 3
[3,2]
 => [2,2,1]
 => 001 => 001 => 1
[3,1,1]
 => [3,1,1]
 => 111 => 111 => 3
[2,2,1]
 => [3,2]
 => 10 => 01 => 1
[2,1,1,1]
 => [4,1]
 => 01 => 01 => 1
[1,1,1,1,1]
 => [5]
 => 1 => 1 => 1
[6]
 => [1,1,1,1,1,1]
 => 111111 => 111111 => 6
[5,1]
 => [2,1,1,1,1]
 => 01111 => 01111 => 4
[4,2]
 => [2,2,1,1]
 => 0011 => 0011 => 2
[4,1,1]
 => [3,1,1,1]
 => 1111 => 1111 => 4
[3,3]
 => [2,2,2]
 => 000 => 000 => 0
[3,2,1]
 => [3,2,1]
 => 101 => 011 => 2
[3,1,1,1]
 => [4,1,1]
 => 011 => 011 => 2
[2,2,2]
 => [3,3]
 => 11 => 11 => 2
[2,2,1,1]
 => [4,2]
 => 00 => 00 => 0
[2,1,1,1,1]
 => [5,1]
 => 11 => 11 => 2
[1,1,1,1,1,1]
 => [6]
 => 0 => 0 => 0
[7]
 => [1,1,1,1,1,1,1]
 => 1111111 => 1111111 => 7
[6,1]
 => [2,1,1,1,1,1]
 => 011111 => 011111 => 5
[5,2]
 => [2,2,1,1,1]
 => 00111 => 00111 => 3
[5,1,1]
 => [3,1,1,1,1]
 => 11111 => 11111 => 5
[4,3]
 => [2,2,2,1]
 => 0001 => 0001 => 1
[4,2,1]
 => [3,2,1,1]
 => 1011 => 0111 => 3
[4,1,1,1]
 => [4,1,1,1]
 => 0111 => 0111 => 3
[3,3,1]
 => [3,2,2]
 => 100 => 001 => 1
[3,2,2]
 => [3,3,1]
 => 111 => 111 => 3
[3,2,1,1]
 => [4,2,1]
 => 001 => 001 => 1
[3,1,1,1,1]
 => [5,1,1]
 => 111 => 111 => 3
[2,2,2,1]
 => [4,3]
 => 01 => 01 => 1
[2,2,1,1,1]
 => [5,2]
 => 10 => 01 => 1
[2,1,1,1,1,1]
 => [6,1]
 => 01 => 01 => 1
[1,1,1,1,1,1,1]
 => [7]
 => 1 => 1 => 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000475
Mapping
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%
Values
[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
Description
The number of parts equal to 1 in a partition.
Matching statistic: St001247
Mapping
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%
Values
[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
Description
The number of parts of a partition that are not congruent 2 modulo 3.
Matching statistic: St001249
Mapping
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%
Values
[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
Description
Sum of the odd parts of a partition.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001419The length of the longest palindromic factor beginning with a one of a binary word. St000696The number of cycles in the breakpoint graph of a permutation. St000895The number of ones on the main diagonal of an alternating sign matrix. St000241The number of cyclical small excedances. St000894The trace of an alternating sign matrix. St001903The number of fixed points of a parking function. St000884The number of isolated descents of a permutation. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.