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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St000992
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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: St000148
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
St000148: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1
[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
[8]
=> [1,1,1,1,1,1,1,1]
=> 8
[7,1]
=> [2,1,1,1,1,1,1]
=> 6
[6,2]
=> [2,2,1,1,1,1]
=> 4
[6,1,1]
=> [3,1,1,1,1,1]
=> 6
[5,3]
=> [2,2,2,1,1]
=> 2
[5,2,1]
=> [3,2,1,1,1]
=> 4
[6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 3
[7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? = 4
[7,5,4,3,1]
=> [5,4,4,3,2,1,1]
=> ? = 4
[11,7,5,1]
=> [4,3,3,3,3,2,2,1,1,1,1]
=> ? = 8
[9,7,5,3,1]
=> [5,4,4,3,3,2,2,1,1]
=> ? = 5
[11,9,7,5,3,1]
=> [6,5,5,4,4,3,3,2,2,1,1]
=> ? = 6
[9,7,5,5,3,1]
=> [6,5,5,4,4,2,2,1,1]
=> ? = 4
[11,8,7,5,4,1]
=> [6,5,5,5,4,3,3,2,1,1,1]
=> ? = 8
[8,7,6,5,4,3,2,1]
=> [8,7,6,5,4,3,2,1]
=> ? = 4
Description
The number of odd parts of a partition.
Matching statistic: St000288
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 95%●distinct values known / distinct values provided: 85%
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 85% ●values known / values provided: 95%●distinct values known / distinct values provided: 85%
Values
[1]
=> [1]
=> 1 => 1 => 1
[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
[8]
=> [1,1,1,1,1,1,1,1]
=> 11111111 => 11111111 => 8
[7,1]
=> [2,1,1,1,1,1,1]
=> 0111111 => 0111111 => 6
[6,2]
=> [2,2,1,1,1,1]
=> 001111 => 001111 => 4
[6,1,1]
=> [3,1,1,1,1,1]
=> 111111 => 111111 => 6
[5,3]
=> [2,2,2,1,1]
=> 00011 => 00011 => 2
[5,2,1]
=> [3,2,1,1,1]
=> 10111 => 01111 => 4
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => ? = 10
[11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111 => ? => ? = 11
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 111111111111 => 111111111111 => ? = 12
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 0011111111 => ? => ? = 8
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> 1111111111 => 1111111111 => ? = 10
[]
=> []
=> ? => ? => ? = 0
[6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? => ? => ? = 3
[7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? => ? => ? = 4
[7,5,4,3,1]
=> [5,4,4,3,2,1,1]
=> ? => ? => ? = 4
[11,7,5,1]
=> [4,3,3,3,3,2,2,1,1,1,1]
=> ? => ? => ? = 8
[9,7,5,3,1]
=> [5,4,4,3,3,2,2,1,1]
=> ? => ? => ? = 5
[11,9,7,5,3,1]
=> [6,5,5,4,4,3,3,2,2,1,1]
=> ? => ? => ? = 6
[9,7,5,5,3,1]
=> [6,5,5,4,4,2,2,1,1]
=> ? => ? => ? = 4
[11,8,7,5,4,1]
=> [6,5,5,5,4,3,3,2,1,1,1]
=> ? => ? => ? = 8
[8,7,6,5,4,3,2,1]
=> [8,7,6,5,4,3,2,1]
=> ? => ? => ? = 4
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
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: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 92%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 92%
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
[11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ?
=> ? = 11
[10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ?
=> ? = 9
[9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ?
=> ? = 7
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ?
=> ? = 9
[8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ?
=> ? = 5
[8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? => ?
=> ? = 7
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ?
=> ? = 7
[7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ?
=> ? = 3
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? => ?
=> ? = 5
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? => ?
=> ? = 7
[7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ? => ?
=> ? = 5
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? => ?
=> ? = 7
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ?
=> ? = 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ?
=> ? = 3
[6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ?
=> ? = 5
[6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ? => ?
=> ? = 3
[6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? => ?
=> ? = 5
[6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? => ?
=> ? = 5
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ?
=> ? = 5
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ?
=> ? = 1
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => ?
=> ? = 3
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6,11] => ?
=> ? = 1
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => ?
=> ? = 5
[5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ?
=> ? = 3
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => ?
=> ? = 3
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10,11] => ?
=> ? = 3
[5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9,10,11] => ?
=> ? = 5
[5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? => ?
=> ? = 3
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ? => ?
=> ? = 5
[4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ?
=> ? = 3
[4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [8,4,9,2,5,10,11,1,3,6,7] => ?
=> ? = 1
[4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> [8,4,3,2,9,10,11,1,5,6,7] => ?
=> ? = 1
[4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11] => ?
=> ? = 3
[4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11] => ?
=> ? = 1
[4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7,11] => ?
=> ? = 3
[4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ?
=> ? = 1
[4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10,11] => ?
=> ? = 3
[4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ? => ?
=> ? = 3
[4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ? => ?
=> ? = 3
[4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? => ?
=> ? = 3
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ?
=> ? = 1
[3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ?
=> ? = 3
[3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8] => ?
=> ? = 1
[3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ?
=> ? = 1
[3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ?
=> ? = 1
[3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ?
=> ? = 3
[3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ?
=> ? = 1
[3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ? => ?
=> ? = 3
[3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? => ?
=> ? = 1
[3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? => ?
=> ? = 3
Description
The number of parts equal to 1 in a partition.
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: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 92%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001247: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 92%
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
[11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ?
=> ? = 11
[10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ?
=> ? = 9
[9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ?
=> ? = 7
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ?
=> ? = 9
[8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ?
=> ? = 5
[8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? => ?
=> ? = 7
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ?
=> ? = 7
[7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ?
=> ? = 3
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? => ?
=> ? = 5
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? => ?
=> ? = 7
[7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ? => ?
=> ? = 5
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? => ?
=> ? = 7
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ?
=> ? = 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ?
=> ? = 3
[6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ?
=> ? = 5
[6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ? => ?
=> ? = 3
[6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? => ?
=> ? = 5
[6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? => ?
=> ? = 5
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ?
=> ? = 5
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ?
=> ? = 1
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => ?
=> ? = 3
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6,11] => ?
=> ? = 1
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => ?
=> ? = 5
[5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ?
=> ? = 3
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => ?
=> ? = 3
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10,11] => ?
=> ? = 3
[5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9,10,11] => ?
=> ? = 5
[5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? => ?
=> ? = 3
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ? => ?
=> ? = 5
[4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ?
=> ? = 3
[4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [8,4,9,2,5,10,11,1,3,6,7] => ?
=> ? = 1
[4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> [8,4,3,2,9,10,11,1,5,6,7] => ?
=> ? = 1
[4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11] => ?
=> ? = 3
[4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11] => ?
=> ? = 1
[4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7,11] => ?
=> ? = 3
[4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ?
=> ? = 1
[4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10,11] => ?
=> ? = 3
[4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ? => ?
=> ? = 3
[4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ? => ?
=> ? = 3
[4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? => ?
=> ? = 3
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ?
=> ? = 1
[3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ?
=> ? = 3
[3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8] => ?
=> ? = 1
[3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ?
=> ? = 1
[3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ?
=> ? = 1
[3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ?
=> ? = 3
[3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ?
=> ? = 1
[3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ? => ?
=> ? = 3
[3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? => ?
=> ? = 1
[3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? => ?
=> ? = 3
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: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 92%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St001249: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 92%
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
[11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ?
=> ? = 11
[10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ?
=> ? = 9
[9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ?
=> ? = 7
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ?
=> ? = 9
[8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ?
=> ? = 5
[8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? => ?
=> ? = 7
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ?
=> ? = 7
[7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ?
=> ? = 3
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? => ?
=> ? = 5
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? => ?
=> ? = 7
[7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ? => ?
=> ? = 5
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? => ?
=> ? = 7
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ?
=> ? = 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ?
=> ? = 3
[6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ?
=> ? = 5
[6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ? => ?
=> ? = 3
[6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? => ?
=> ? = 5
[6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? => ?
=> ? = 5
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ?
=> ? = 5
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ?
=> ? = 1
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => ?
=> ? = 3
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6,11] => ?
=> ? = 1
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => ?
=> ? = 5
[5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ?
=> ? = 3
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => ?
=> ? = 3
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10,11] => ?
=> ? = 3
[5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9,10,11] => ?
=> ? = 5
[5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? => ?
=> ? = 3
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ? => ?
=> ? = 5
[4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ?
=> ? = 3
[4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [8,4,9,2,5,10,11,1,3,6,7] => ?
=> ? = 1
[4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> [8,4,3,2,9,10,11,1,5,6,7] => ?
=> ? = 1
[4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11] => ?
=> ? = 3
[4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11] => ?
=> ? = 1
[4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7,11] => ?
=> ? = 3
[4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ?
=> ? = 1
[4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10,11] => ?
=> ? = 3
[4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ? => ?
=> ? = 3
[4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ? => ?
=> ? = 3
[4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? => ?
=> ? = 3
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ?
=> ? = 1
[3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ?
=> ? = 3
[3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8] => ?
=> ? = 1
[3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ?
=> ? = 1
[3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ?
=> ? = 1
[3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ?
=> ? = 3
[3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ?
=> ? = 1
[3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ? => ?
=> ? = 3
[3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? => ?
=> ? = 1
[3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? => ?
=> ? = 3
Description
Sum of the odd parts of a partition.
Matching statistic: St000022
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 85%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 85%
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
[11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 11
[10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? = 9
[9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? = 7
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ? = 9
[8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 5
[8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? => ? = 7
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? = 7
[7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ? = 3
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? => ? = 5
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? => ? = 7
[7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ? => ? = 5
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? => ? = 7
[6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 3
[6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 5
[6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ? => ? = 3
[6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? => ? = 5
[6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? => ? = 5
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? = 5
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? = 1
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => ? = 3
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6,11] => ? = 1
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => ? = 5
[5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ? = 3
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => ? = 3
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10,11] => ? = 3
[5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9,10,11] => ? = 5
[5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? => ? = 3
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ? => ? = 5
[4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? = 3
[4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [8,4,9,2,5,10,11,1,3,6,7] => ? = 1
[4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> [8,4,3,2,9,10,11,1,5,6,7] => ? = 1
[4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11] => ? = 3
[4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11] => ? = 1
[4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7,11] => ? = 3
[4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ? = 1
[4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10,11] => ? = 3
[4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ? => ? = 3
[4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ? => ? = 3
[4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? => ? = 3
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? = 1
[3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ? = 3
[3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8] => ? = 1
[3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
[3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ? = 1
[3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 3
[3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
[3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ? => ? = 3
[3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? => ? = 1
[3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? => ? = 3
Description
The number of fixed points of a permutation.
Matching statistic: St000247
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: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 62%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 62%
Values
[1]
=> [[1]]
=> [1] => {{1}}
=> ? = 1
[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
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [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] => {{1,2},{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] => {{1,3},{2,4},{5},{6},{7},{8}}
=> ? = 4
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 6
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => {{1,4},{2,5},{3,6},{7},{8}}
=> ? = 2
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => {{1,4},{2},{3,5},{6},{7},{8}}
=> ? = 4
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => {{1,4},{2,3},{5},{6},{7},{8}}
=> ? = 4
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => {{1,5},{2,6},{3,7},{4,8}}
=> ? = 0
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => {{1,5},{2},{3,6},{4,7},{8}}
=> ? = 2
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => {{1,5},{2,6},{3},{4},{7},{8}}
=> ? = 4
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => {{1,5},{2,3},{4,6},{7},{8}}
=> ? = 2
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => {{1,5},{2,4},{3},{6},{7},{8}}
=> ? = 4
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => {{1,6},{2,7},{3},{4},{5,8}}
=> ? = 2
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => {{1,6},{2,3},{4,7},{5,8}}
=> ? = 0
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => {{1,6},{2,4},{3,7},{5},{8}}
=> ? = 2
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => {{1,6},{2,4},{3},{5,7},{8}}
=> ? = 2
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => {{1,6},{2,5},{3,4},{7},{8}}
=> ? = 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => {{1,7},{2,8},{3,5},{4,6}}
=> ? = 0
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => {{1,7},{2,5},{3},{4,8},{6}}
=> ? = 2
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => {{1,7},{2,5},{3,4},{6,8}}
=> ? = 0
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => {{1,7},{2,6},{3,5},{4},{8}}
=> ? = 2
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => {{1,8},{2,7},{3,6},{4,5}}
=> ? = 0
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 9
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => {{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 7
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => {{1,3},{2,4},{5},{6},{7},{8},{9}}
=> ? = 5
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => {{1,3},{2},{4},{5},{6},{7},{8},{9}}
=> ? = 7
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => {{1,4},{2,5},{3,6},{7},{8},{9}}
=> ? = 3
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => {{1,4},{2},{3,5},{6},{7},{8},{9}}
=> ? = 5
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => {{1,4},{2,3},{5},{6},{7},{8},{9}}
=> ? = 5
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => {{1,5},{2,6},{3,7},{4,8},{9}}
=> ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => {{1,5},{2},{3,6},{4,7},{8},{9}}
=> ? = 3
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => {{1,5},{2,6},{3},{4},{7},{8},{9}}
=> ? = 5
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => {{1,5},{2,3},{4,6},{7},{8},{9}}
=> ? = 3
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => {{1,5},{2,4},{3},{6},{7},{8},{9}}
=> ? = 5
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => {{1,6},{2},{3,7},{4,8},{5,9}}
=> ? = 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => {{1,6},{2,7},{3},{4},{5,8},{9}}
=> ? = 3
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => {{1,6},{2,3},{4,7},{5,8},{9}}
=> ? = 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => {{1,6},{2,4},{3,7},{5},{8},{9}}
=> ? = 3
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => {{1,6},{2,4},{3},{5,7},{8},{9}}
=> ? = 3
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => {{1,6},{2,5},{3,4},{7},{8},{9}}
=> ? = 3
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => {{1,7},{2,8},{3,9},{4},{5},{6}}
=> ? = 3
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => {{1,7},{2,4},{3,8},{5},{6,9}}
=> ? = 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => {{1,7},{2,4},{3},{5,8},{6,9}}
=> ? = 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => {{1,7},{2,8},{3,5},{4,6},{9}}
=> ? = 1
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => {{1,7},{2,5},{3},{4,8},{6},{9}}
=> ? = 3
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => {{1,7},{2,5},{3,4},{6,8},{9}}
=> ? = 1
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => {{1,7},{2,6},{3,5},{4},{8},{9}}
=> ? = 3
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3] => {{1,8},{2,6},{3,9},{4},{5,7}}
=> ? = 1
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> [8,6,4,3,9,2,7,1,5] => {{1,8},{2,6},{3,4},{5,9},{7}}
=> ? = 1
Description
The number of singleton blocks of a set partition.
Matching statistic: St000696
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000696: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 85%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000696: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 85%
Values
[1]
=> [[1]]
=> [1] => [1] => 2 = 1 + 1
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 3 = 2 + 1
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1 = 0 + 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 4 = 3 + 1
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 2 = 1 + 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 5 = 4 + 1
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => 1 = 0 + 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => 3 = 2 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => 1 = 0 + 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 4 = 3 + 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => 2 = 1 + 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => 4 = 3 + 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 2 = 1 + 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [3,2,4,1,5] => 2 = 1 + 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [3,4,2,5,1] => 2 = 1 + 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 7 = 6 + 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 5 = 4 + 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => 3 = 2 + 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => 5 = 4 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => 1 = 0 + 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,4,1,5,3,6] => 3 = 2 + 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,2,4,1,5,6] => 3 = 2 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [3,4,5,1,6,2] => 3 = 2 + 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,2,5,1,6,4] => 1 = 0 + 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [3,4,2,5,1,6] => 3 = 2 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => 1 = 0 + 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 8 = 7 + 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 6 = 5 + 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [3,1,4,2,5,6,7] => ? = 3 + 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => 6 = 5 + 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [4,1,5,2,6,3,7] => ? = 1 + 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => ? = 3 + 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [3,2,4,1,5,6,7] => ? = 3 + 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,5,1,6,3,7,4] => ? = 1 + 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [3,4,5,1,6,2,7] => ? = 3 + 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [3,2,5,1,6,4,7] => ? = 1 + 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [3,4,2,5,1,6,7] => ? = 3 + 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [4,2,5,6,1,7,3] => ? = 1 + 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [3,4,2,6,1,7,5] => ? = 1 + 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [4,3,5,2,6,1,7] => ? = 1 + 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 1 + 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 9 = 8 + 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 7 = 6 + 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [3,1,4,2,5,6,7,8] => ? = 4 + 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [2,3,1,4,5,6,7,8] => ? = 6 + 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [4,1,5,2,6,3,7,8] => ? = 2 + 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [2,4,1,5,3,6,7,8] => ? = 4 + 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [3,2,4,1,5,6,7,8] => ? = 4 + 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => 1 = 0 + 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [2,5,1,6,3,7,4,8] => ? = 2 + 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [3,4,5,1,6,2,7,8] => ? = 4 + 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [3,2,5,1,6,4,7,8] => ? = 2 + 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [3,4,2,5,1,6,7,8] => ? = 4 + 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [3,4,6,1,7,2,8,5] => ? = 2 + 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [3,2,6,1,7,4,8,5] => 1 = 0 + 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [4,2,5,6,1,7,3,8] => ? = 2 + 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [3,4,2,6,1,7,5,8] => ? = 2 + 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [4,3,5,2,6,1,7,8] => ? = 2 + 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [5,3,6,4,7,1,8,2] => 1 = 0 + 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [3,5,2,6,7,1,8,4] => ? = 2 + 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [4,3,5,2,7,1,8,6] => 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [4,5,3,6,2,7,1,8] => ? = 2 + 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => 1 = 0 + 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 10 = 9 + 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 8 = 7 + 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [3,1,4,2,5,6,7,8,9] => ? = 5 + 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [2,3,1,4,5,6,7,8,9] => ? = 7 + 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [4,1,5,2,6,3,7,8,9] => ? = 3 + 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [2,4,1,5,3,6,7,8,9] => ? = 5 + 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [3,2,4,1,5,6,7,8,9] => ? = 5 + 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [5,1,6,2,7,3,8,4,9] => ? = 1 + 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [2,5,1,6,3,7,4,8,9] => ? = 3 + 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [3,4,5,1,6,2,7,8,9] => ? = 5 + 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [3,2,5,1,6,4,7,8,9] => ? = 3 + 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [3,4,2,5,1,6,7,8,9] => ? = 5 + 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [2,6,1,7,3,8,4,9,5] => ? = 1 + 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [3,4,6,1,7,2,8,5,9] => ? = 3 + 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [3,2,6,1,7,4,8,5,9] => ? = 1 + 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [4,2,5,6,1,7,3,8,9] => ? = 3 + 1
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [3,4,2,6,1,7,5,8,9] => ? = 3 + 1
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [4,3,5,2,6,1,7,8,9] => ? = 3 + 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [4,5,6,7,1,8,2,9,3] => ? = 3 + 1
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [4,2,5,7,1,8,3,9,6] => ? = 1 + 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [3,4,2,7,1,8,5,9,6] => ? = 1 + 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => [5,3,6,4,7,1,8,2,9] => ? = 1 + 1
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => [3,5,2,6,7,1,8,4,9] => ? = 3 + 1
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => [4,3,5,2,7,1,8,6,9] => ? = 1 + 1
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => [4,5,3,6,2,7,1,8,9] => ? = 3 + 1
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 11 = 10 + 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => 9 = 8 + 1
Description
The number of cycles in the breakpoint graph of a permutation.
The breakpoint graph of a permutation $\pi_1,\dots,\pi_n$ is the directed, bicoloured graph with vertices $0,\dots,n$, a grey edge from $i$ to $i+1$ and a black edge from $\pi_i$ to $\pi_{i-1}$ for $0\leq i\leq n$, all indices taken modulo $n+1$.
This graph decomposes into alternating cycles, which this statistic counts.
The distribution of this statistic on permutations of $n-1$ is, according to [cor.1, 5] and [eq.6, 6], given by
$$
\frac{1}{n(n+1)}((q+n)_{n+1}-(q)_{n+1}),
$$
where $(x)_n=x(x-1)\dots(x-n+1)$.
Matching statistic: St000895
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 54%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 54%
Values
[1]
=> [[1]]
=> [1] => [[1]]
=> 1
[2]
=> [[1,2]]
=> [1,2] => [[1,0],[0,1]]
=> 2
[1,1]
=> [[1],[2]]
=> [2,1] => [[0,1],[1,0]]
=> 0
[3]
=> [[1,2,3]]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 1
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 0
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 0
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 3
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
=> 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
=> 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 1
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 6
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 4
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 4
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> 2
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> 0
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> 2
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> 0
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 7
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 5
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 5
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [[0,0,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [[0,0,0,0,1,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0]]
=> ? = 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 3
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[1,0,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 3
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[1,0,0,0,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,0,1],[1,0,0,0,0,0,0],[0,0,0,0,1,0,0]]
=> ? = 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0],[0,0,0,0,0,0,1]]
=> 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [[0,0,0,0,0,0,1],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[1,0,0,0,0,0,0]]
=> ? = 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 8
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 6
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 4
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 6
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 2
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 4
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 4
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0]]
=> 0
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [[0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> ? = 2
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 4
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 2
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 4
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0]]
=> ? = 2
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0]]
=> 0
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> ? = 2
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [[0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1]]
=> ? = 2
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? = 2
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0]]
=> 0
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0]]
=> ? = 2
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [[0,0,0,0,0,0,1,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0]]
=> 0
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,1]]
=> 2
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0]]
=> 0
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 9
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 7
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 5
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 7
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 3
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [[0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 5
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 5
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [[0,0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 3
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 5
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [[0,0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 3
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 5
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [[0,0,0,0,0,1,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0]]
=> ? = 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 3
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [[0,0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [[0,0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 3
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [[0,0,0,0,0,1,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 3
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? = 3
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0]]
=> ? = 3
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [[0,0,0,0,0,0,1,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1,0],[0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0]]
=> ? = 1
[2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9] => [[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,1]]
=> 1
[2,1,1,1,1,1,1,1,1]
=> [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1,10] => [[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,1]]
=> 2
Description
The number of ones on the main diagonal of an alternating sign matrix.
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