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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000992
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000992: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> 0
[2]
=> []
=> 0
[1,1]
=> [1]
=> 1
[3]
=> []
=> 0
[2,1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> 0
[4]
=> []
=> 0
[3,1]
=> [1]
=> 1
[2,2]
=> [2]
=> 2
[2,1,1]
=> [1,1]
=> 0
[1,1,1,1]
=> [1,1,1]
=> 1
[5]
=> []
=> 0
[4,1]
=> [1]
=> 1
[3,2]
=> [2]
=> 2
[3,1,1]
=> [1,1]
=> 0
[2,2,1]
=> [2,1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 0
[6]
=> []
=> 0
[5,1]
=> [1]
=> 1
[4,2]
=> [2]
=> 2
[4,1,1]
=> [1,1]
=> 0
[3,3]
=> [3]
=> 3
[3,2,1]
=> [2,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> 1
[2,2,2]
=> [2,2]
=> 0
[2,2,1,1]
=> [2,1,1]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[7]
=> []
=> 0
[6,1]
=> [1]
=> 1
[5,2]
=> [2]
=> 2
[5,1,1]
=> [1,1]
=> 0
[4,3]
=> [3]
=> 3
[4,2,1]
=> [2,1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> 1
[3,3,1]
=> [3,1]
=> 2
[3,2,2]
=> [2,2]
=> 0
[3,2,1,1]
=> [2,1,1]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 0
[8]
=> []
=> 0
[7,1]
=> [1]
=> 1
[6,2]
=> [2]
=> 2
[6,1,1]
=> [1,1]
=> 0
[5,3]
=> [3]
=> 3
[5,2,1]
=> [2,1]
=> 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 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> 0
[2]
=> []
=> []
=> 0
[1,1]
=> [1]
=> [1]
=> 1
[3]
=> []
=> []
=> 0
[2,1]
=> [1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> 0
[4]
=> []
=> []
=> 0
[3,1]
=> [1]
=> [1]
=> 1
[2,2]
=> [2]
=> [1,1]
=> 2
[2,1,1]
=> [1,1]
=> [2]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[5]
=> []
=> []
=> 0
[4,1]
=> [1]
=> [1]
=> 1
[3,2]
=> [2]
=> [1,1]
=> 2
[3,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[6]
=> []
=> []
=> 0
[5,1]
=> [1]
=> [1]
=> 1
[4,2]
=> [2]
=> [1,1]
=> 2
[4,1,1]
=> [1,1]
=> [2]
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> 3
[3,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[2,2,2]
=> [2,2]
=> [2,2]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[7]
=> []
=> []
=> 0
[6,1]
=> [1]
=> [1]
=> 1
[5,2]
=> [2]
=> [1,1]
=> 2
[5,1,1]
=> [1,1]
=> [2]
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> 3
[4,2,1]
=> [2,1]
=> [2,1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,2,2]
=> [2,2]
=> [2,2]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0
[8]
=> []
=> []
=> 0
[7,1]
=> [1]
=> [1]
=> 1
[6,2]
=> [2]
=> [1,1]
=> 2
[6,1,1]
=> [1,1]
=> [2]
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> 3
[5,2,1]
=> [2,1]
=> [2,1]
=> 1
[7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 3
[6,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 3
[7,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? = 2
[6,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? = 2
[5,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? = 2
[7,6,4,4,3,2,1]
=> [6,4,4,3,2,1]
=> [6,5,4,3,1,1]
=> ? = 4
[6,6,4,4,3,2,1]
=> [6,4,4,3,2,1]
=> [6,5,4,3,1,1]
=> ? = 4
[7,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> [6,5,4,3,1]
=> ? = 3
[6,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> [6,5,4,3,1]
=> ? = 3
[5,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> [6,5,4,3,1]
=> ? = 3
[7,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? = 2
[6,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? = 2
[5,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? = 2
[4,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? = 2
[6,6,5,3,3,2,1]
=> [6,5,3,3,2,1]
=> [6,5,4,2,2,1]
=> ? = 2
[6,5,4,3,3,2,1]
=> [5,4,3,3,2,1]
=> [6,5,4,2,1]
=> ? = 2
[6,6,5,4,2,2,1]
=> [6,5,4,2,2,1]
=> [6,5,3,3,2,1]
=> ? = 4
[6,5,5,4,2,2,1]
=> [5,5,4,2,2,1]
=> [6,5,3,3,2]
=> ? = 3
[7,6,5,2,2,2,1]
=> [6,5,2,2,2,1]
=> [6,5,2,2,2,1]
=> ? = 2
[7,6,5,4,3,1,1]
=> [6,5,4,3,1,1]
=> [6,4,4,3,2,1]
=> ? = 2
[6,6,5,4,3,1,1]
=> [6,5,4,3,1,1]
=> [6,4,4,3,2,1]
=> ? = 2
[6,5,5,4,3,1,1]
=> [5,5,4,3,1,1]
=> [6,4,4,3,2]
=> ? = 1
[7,6,5,4,1,1,1]
=> [6,5,4,1,1,1]
=> [6,3,3,3,2,1]
=> ? = 4
[7,6,5,4,3,2]
=> [6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? = 4
[6,6,5,4,3,2]
=> [6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? = 4
[7,5,5,4,3,2]
=> [5,5,4,3,2]
=> [5,5,4,3,2]
=> ? = 3
[6,5,5,4,3,2]
=> [5,5,4,3,2]
=> [5,5,4,3,2]
=> ? = 3
[5,5,5,4,3,2]
=> [5,5,4,3,2]
=> [5,5,4,3,2]
=> ? = 3
[7,6,4,4,3,2]
=> [6,4,4,3,2]
=> [5,5,4,3,1,1]
=> ? = 5
[7,6,5,2,2,2]
=> [6,5,2,2,2]
=> [5,5,2,2,2,1]
=> ? = 3
[7,6,5,4,3,1]
=> [6,5,4,3,1]
=> [5,4,4,3,2,1]
=> ? = 3
[6,6,5,4,3,1]
=> [6,5,4,3,1]
=> [5,4,4,3,2,1]
=> ? = 3
[7,6,5,4,3]
=> [6,5,4,3]
=> [4,4,4,3,2,1]
=> ? = 2
[6,6,5,4,3]
=> [6,5,4,3]
=> [4,4,4,3,2,1]
=> ? = 2
[6,5,5,4,3]
=> [5,5,4,3]
=> [4,4,4,3,2]
=> ? = 1
[3,3,3,3,3,3,3,3]
=> [3,3,3,3,3,3,3]
=> [7,7,7]
=> ? = 3
[2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2]
=> [9,9]
=> ? = 2
[3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> [6,6,6]
=> ? = 0
[4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [5,5,5,5]
=> ? = 4
[2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
=> [15,15]
=> ? = 2
[6,6,6,6]
=> [6,6,6]
=> [3,3,3,3,3,3]
=> ? = 6
[6,6,6,6,6]
=> [6,6,6,6]
=> [4,4,4,4,4,4]
=> ? = 0
[10,10,10,10]
=> [10,10,10]
=> [3,3,3,3,3,3,3,3,3,3]
=> ? = 10
[5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]
=> [23,23,23,23,23]
=> ? = 5
[5,5,5,5,5,5,5,5,5,5,5,5]
=> [5,5,5,5,5,5,5,5,5,5,5]
=> [11,11,11,11,11]
=> ? = 5
[5,5,5,5,5,5]
=> [5,5,5,5,5]
=> [5,5,5,5,5]
=> ? = 5
[11,9,7,5,3,1]
=> [9,7,5,3,1]
=> [5,4,4,3,3,2,2,1,1]
=> ? = 5
[8,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? = 4
[7,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? = 4
[7,6,6,5,4,3,2,1]
=> [6,6,5,4,3,2,1]
=> [7,6,5,4,3,2]
=> ? = 3
Description
The number of odd parts of a partition.
Matching statistic: St000288
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 93%●distinct values known / distinct values provided: 77%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 93%●distinct values known / distinct values provided: 77%
Values
[1]
=> []
=> []
=> ? => ? = 0
[2]
=> []
=> []
=> ? => ? = 0
[1,1]
=> [1]
=> [1]
=> 1 => 1
[3]
=> []
=> []
=> ? => ? = 0
[2,1]
=> [1]
=> [1]
=> 1 => 1
[1,1,1]
=> [1,1]
=> [2]
=> 0 => 0
[4]
=> []
=> []
=> ? => ? = 0
[3,1]
=> [1]
=> [1]
=> 1 => 1
[2,2]
=> [2]
=> [1,1]
=> 11 => 2
[2,1,1]
=> [1,1]
=> [2]
=> 0 => 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 1 => 1
[5]
=> []
=> []
=> ? => ? = 0
[4,1]
=> [1]
=> [1]
=> 1 => 1
[3,2]
=> [2]
=> [1,1]
=> 11 => 2
[3,1,1]
=> [1,1]
=> [2]
=> 0 => 0
[2,2,1]
=> [2,1]
=> [2,1]
=> 01 => 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 1 => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 => 0
[6]
=> []
=> []
=> ? => ? = 0
[5,1]
=> [1]
=> [1]
=> 1 => 1
[4,2]
=> [2]
=> [1,1]
=> 11 => 2
[4,1,1]
=> [1,1]
=> [2]
=> 0 => 0
[3,3]
=> [3]
=> [1,1,1]
=> 111 => 3
[3,2,1]
=> [2,1]
=> [2,1]
=> 01 => 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 1 => 1
[2,2,2]
=> [2,2]
=> [2,2]
=> 00 => 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 11 => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1 => 1
[7]
=> []
=> []
=> ? => ? = 0
[6,1]
=> [1]
=> [1]
=> 1 => 1
[5,2]
=> [2]
=> [1,1]
=> 11 => 2
[5,1,1]
=> [1,1]
=> [2]
=> 0 => 0
[4,3]
=> [3]
=> [1,1,1]
=> 111 => 3
[4,2,1]
=> [2,1]
=> [2,1]
=> 01 => 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 1 => 1
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 011 => 2
[3,2,2]
=> [2,2]
=> [2,2]
=> 00 => 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 11 => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 => 0
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 10 => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 01 => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1 => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0 => 0
[8]
=> []
=> []
=> ? => ? = 0
[7,1]
=> [1]
=> [1]
=> 1 => 1
[6,2]
=> [2]
=> [1,1]
=> 11 => 2
[6,1,1]
=> [1,1]
=> [2]
=> 0 => 0
[5,3]
=> [3]
=> [1,1,1]
=> 111 => 3
[5,2,1]
=> [2,1]
=> [2,1]
=> 01 => 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> 1 => 1
[4,4]
=> [4]
=> [1,1,1,1]
=> 1111 => 4
[4,3,1]
=> [3,1]
=> [2,1,1]
=> 011 => 2
[4,2,2]
=> [2,2]
=> [2,2]
=> 00 => 0
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 11 => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0 => 0
[3,3,2]
=> [3,2]
=> [2,2,1]
=> 001 => 1
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 111 => 3
[9]
=> []
=> []
=> ? => ? = 0
[10]
=> []
=> []
=> ? => ? = 0
[11]
=> []
=> []
=> ? => ? = 0
[12]
=> []
=> []
=> ? => ? = 0
[13]
=> []
=> []
=> ? => ? = 0
[14]
=> []
=> []
=> ? => ? = 0
[15]
=> []
=> []
=> ? => ? = 0
[16]
=> []
=> []
=> ? => ? = 0
[17]
=> []
=> []
=> ? => ? = 0
[7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? => ? = 3
[6,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? => ? = 3
[7,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? => ? = 2
[6,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? => ? = 2
[5,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? => ? = 2
[7,6,4,4,3,2,1]
=> [6,4,4,3,2,1]
=> [6,5,4,3,1,1]
=> ? => ? = 4
[6,6,4,4,3,2,1]
=> [6,4,4,3,2,1]
=> [6,5,4,3,1,1]
=> ? => ? = 4
[7,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> [6,5,4,3,1]
=> ? => ? = 3
[6,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> [6,5,4,3,1]
=> ? => ? = 3
[5,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> [6,5,4,3,1]
=> ? => ? = 3
[7,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? => ? = 2
[6,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? => ? = 2
[5,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? => ? = 2
[4,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? => ? = 2
[6,6,5,3,3,2,1]
=> [6,5,3,3,2,1]
=> [6,5,4,2,2,1]
=> ? => ? = 2
[6,5,4,3,3,2,1]
=> [5,4,3,3,2,1]
=> [6,5,4,2,1]
=> ? => ? = 2
[6,6,5,4,2,2,1]
=> [6,5,4,2,2,1]
=> [6,5,3,3,2,1]
=> ? => ? = 4
[6,5,5,4,2,2,1]
=> [5,5,4,2,2,1]
=> [6,5,3,3,2]
=> ? => ? = 3
[7,6,5,2,2,2,1]
=> [6,5,2,2,2,1]
=> [6,5,2,2,2,1]
=> ? => ? = 2
[7,6,5,4,3,1,1]
=> [6,5,4,3,1,1]
=> [6,4,4,3,2,1]
=> ? => ? = 2
[6,6,5,4,3,1,1]
=> [6,5,4,3,1,1]
=> [6,4,4,3,2,1]
=> ? => ? = 2
[6,5,5,4,3,1,1]
=> [5,5,4,3,1,1]
=> [6,4,4,3,2]
=> ? => ? = 1
[7,6,5,4,1,1,1]
=> [6,5,4,1,1,1]
=> [6,3,3,3,2,1]
=> ? => ? = 4
[7,6,5,4,3,2]
=> [6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? => ? = 4
[6,6,5,4,3,2]
=> [6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? => ? = 4
[7,5,5,4,3,2]
=> [5,5,4,3,2]
=> [5,5,4,3,2]
=> ? => ? = 3
[6,5,5,4,3,2]
=> [5,5,4,3,2]
=> [5,5,4,3,2]
=> ? => ? = 3
[5,5,5,4,3,2]
=> [5,5,4,3,2]
=> [5,5,4,3,2]
=> ? => ? = 3
[7,6,4,4,3,2]
=> [6,4,4,3,2]
=> [5,5,4,3,1,1]
=> ? => ? = 5
[7,6,5,4,3,1]
=> [6,5,4,3,1]
=> [5,4,4,3,2,1]
=> ? => ? = 3
[6,6,5,4,3,1]
=> [6,5,4,3,1]
=> [5,4,4,3,2,1]
=> ? => ? = 3
[7,6,5,4,3]
=> [6,5,4,3]
=> [4,4,4,3,2,1]
=> ? => ? = 2
[6,6,5,4,3]
=> [6,5,4,3]
=> [4,4,4,3,2,1]
=> ? => ? = 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000022
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 85%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 85%
Values
[1]
=> []
=> []
=> [] => 0
[2]
=> []
=> []
=> [] => 0
[1,1]
=> [1]
=> [[1]]
=> [1] => 1
[3]
=> []
=> []
=> [] => 0
[2,1]
=> [1]
=> [[1]]
=> [1] => 1
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4]
=> []
=> []
=> [] => 0
[3,1]
=> [1]
=> [[1]]
=> [1] => 1
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 2
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[5]
=> []
=> []
=> [] => 0
[4,1]
=> [1]
=> [[1]]
=> [1] => 1
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 2
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[6]
=> []
=> []
=> [] => 0
[5,1]
=> [1]
=> [[1]]
=> [1] => 1
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 2
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[7]
=> []
=> []
=> [] => 0
[6,1]
=> [1]
=> [[1]]
=> [1] => 1
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 2
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 2
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0
[8]
=> []
=> []
=> [] => 0
[7,1]
=> [1]
=> [[1]]
=> [1] => 1
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 2
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? => ? = 1
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? => ? = 1
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ? => ? = 1
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 1
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[3,3,3,3,2]
=> [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
[3,3,3,3,1,1]
=> [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
[3,3,3,2,2,1]
=> [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
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ? = 1
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 3
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ? => ? = 3
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? => ? = 1
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? => ? = 3
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? => ? = 1
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? => ? = 1
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ? => ? = 1
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 1
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? = 0
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ? => ? = 2
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ? => ? = 0
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3,12],[4],[5],[6],[7],[9],[11]]
=> ? => ? = 2
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? => ? = 0
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,12],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 2
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? => ? = 1
[4,4,4,3]
=> [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
[4,4,4,2,1]
=> [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
[4,4,4,1,1,1]
=> [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
[4,4,3,3,1]
=> [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
[4,4,3,2,2]
=> [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
[4,4,3,2,1,1]
=> [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
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ? = 1
[4,4,2,2,2,1]
=> [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
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ? => ? = 3
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ? => ? = 3
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? => ? = 3
[4,3,3,3,2]
=> [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
[4,3,3,3,1,1]
=> [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
[4,3,3,2,2,1]
=> [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
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ? = 1
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 3
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ? => ? = 3
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? => ? = 1
Description
The number of fixed points of a permutation.
Matching statistic: St000150
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 69%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 69%
Values
[1]
=> [1]
=> [1]
=> 0
[2]
=> [1,1]
=> [2]
=> 0
[1,1]
=> [2]
=> [1,1]
=> 1
[3]
=> [1,1,1]
=> [2,1]
=> 0
[2,1]
=> [2,1]
=> [1,1,1]
=> 1
[1,1,1]
=> [3]
=> [3]
=> 0
[4]
=> [1,1,1,1]
=> [4]
=> 0
[3,1]
=> [2,1,1]
=> [2,1,1]
=> 1
[2,2]
=> [2,2]
=> [1,1,1,1]
=> 2
[2,1,1]
=> [3,1]
=> [3,1]
=> 0
[1,1,1,1]
=> [4]
=> [2,2]
=> 1
[5]
=> [1,1,1,1,1]
=> [4,1]
=> 0
[4,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> [1,1,1,1,1]
=> 2
[3,1,1]
=> [3,1,1]
=> [3,2]
=> 0
[2,2,1]
=> [3,2]
=> [3,1,1]
=> 1
[2,1,1,1]
=> [4,1]
=> [2,2,1]
=> 1
[1,1,1,1,1]
=> [5]
=> [5]
=> 0
[6]
=> [1,1,1,1,1,1]
=> [4,2]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [4,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [3,2,1]
=> 0
[3,3]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> 3
[3,2,1]
=> [3,2,1]
=> [3,1,1,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [2,2,2]
=> 1
[2,2,2]
=> [3,3]
=> [6]
=> 0
[2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [5,1]
=> 0
[1,1,1,1,1,1]
=> [6]
=> [3,3]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [4,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1,1,1]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [4,3]
=> 0
[4,3]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 3
[4,2,1]
=> [3,2,1,1]
=> [3,2,1,1]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> [2,2,2,1]
=> 1
[3,3,1]
=> [3,2,2]
=> [3,1,1,1,1]
=> 2
[3,2,2]
=> [3,3,1]
=> [6,1]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [2,2,1,1,1]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [5,2]
=> 0
[2,2,2,1]
=> [4,3]
=> [3,2,2]
=> 1
[2,2,1,1,1]
=> [5,2]
=> [5,1,1]
=> 1
[2,1,1,1,1,1]
=> [6,1]
=> [3,3,1]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> [7]
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [8]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [4,2,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [4,1,1,1,1]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [4,3,1]
=> 0
[5,3]
=> [2,2,2,1,1]
=> [2,1,1,1,1,1,1]
=> 3
[5,2,1]
=> [3,2,1,1,1]
=> [3,2,1,1,1]
=> 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
[3,2,2,1,1,1,1]
=> [7,3,1]
=> [7,3,1]
=> ? = 0
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [4,3,2,1,1,1]
=> ? = 1
[8,2,1,1]
=> [4,2,1,1,1,1,1,1]
=> [4,2,2,2,1,1]
=> ? = 2
[6,3,3]
=> [3,3,3,1,1,1]
=> [6,3,2,1]
=> ? = 0
[6,3,2,1]
=> [4,3,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> ? = 2
[5,3,1,1,1,1]
=> [6,2,2,1,1]
=> [3,3,2,1,1,1,1]
=> ? = 3
[5,2,1,1,1,1,1]
=> [7,2,1,1,1]
=> [7,2,1,1,1]
=> ? = 1
[3,3,1,1,1,1,1,1]
=> [8,2,2]
=> [4,4,1,1,1,1]
=> ? = 3
[12,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [8,2,1,1,1]
=> ? = 1
[11,2]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [8,1,1,1,1,1]
=> ? = 2
[10,3]
=> [2,2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> ? = 3
[10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> [8,2,2,1]
=> ? = 1
[9,4]
=> [2,2,2,2,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> ? = 4
[9,3,1]
=> [3,2,2,1,1,1,1,1,1]
=> [4,3,2,1,1,1,1]
=> ? = 2
[9,2,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> [4,2,2,2,1,1,1]
=> ? = 2
[8,4,1]
=> [3,2,2,2,1,1,1,1]
=> [4,3,1,1,1,1,1,1]
=> ? = 3
[8,3,2]
=> [3,3,2,1,1,1,1,1]
=> [6,4,1,1,1]
=> ? = 1
[8,3,1,1]
=> [4,2,2,1,1,1,1,1]
=> [4,2,2,1,1,1,1,1]
=> ? = 3
[8,2,2,1]
=> [4,3,1,1,1,1,1,1]
=> [4,3,2,2,2]
=> ? = 1
[7,4,2]
=> [3,3,2,2,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
[7,3,2,1]
=> [4,3,2,1,1,1,1]
=> [4,3,2,2,1,1]
=> ? = 2
[7,2,1,1,1,1]
=> [6,2,1,1,1,1,1]
=> [4,3,3,1,1,1]
=> ? = 2
[6,4,3]
=> [3,3,3,2,1,1]
=> [6,3,2,1,1]
=> ? = 1
[6,4,2,1]
=> [4,3,2,2,1,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 3
[6,3,2,2]
=> [4,4,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> ? = 3
[5,5,1,1,1]
=> [5,2,2,2,2]
=> [5,1,1,1,1,1,1,1,1]
=> ? = 4
[5,2,2,2,2]
=> [5,5,1,1,1]
=> [10,2,1]
=> ? = 0
[5,2,1,1,1,1,1,1]
=> [8,2,1,1,1]
=> [4,4,2,1,1,1]
=> ? = 2
[4,4,2,1,1,1]
=> [6,3,2,2]
=> [3,3,3,1,1,1,1]
=> ? = 3
[4,3,2,2,2]
=> [5,5,2,1]
=> [10,1,1,1]
=> ? = 1
[4,3,2,1,1,1,1]
=> [7,3,2,1]
=> [7,3,1,1,1]
=> ? = 1
[4,3,1,1,1,1,1,1]
=> [8,2,2,1]
=> [4,4,1,1,1,1,1]
=> ? = 3
[3,3,3,3,1]
=> [5,4,4]
=> [5,2,2,2,2]
=> ? = 2
[3,3,2,2,1,1,1]
=> [7,4,2]
=> [7,2,2,1,1]
=> ? = 2
[3,3,1,1,1,1,1,1,1]
=> [9,2,2]
=> [9,1,1,1,1]
=> ? = 2
[3,2,2,1,1,1,1,1,1]
=> [9,3,1]
=> [9,3,1]
=> ? = 0
[3,2,1,1,1,1,1,1,1,1]
=> [10,2,1]
=> [5,5,1,1,1]
=> ? = 2
[3,1,1,1,1,1,1,1,1,1,1]
=> [11,1,1]
=> [11,2]
=> ? = 0
[2,2,1,1,1,1,1,1,1,1,1]
=> [11,2]
=> [11,1,1]
=> ? = 1
[13,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [8,4,1,1]
=> ? = 1
[12,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> [8,2,1,1,1,1]
=> ? = 2
[12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [8,3,2,1]
=> ? = 0
[11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> [8,2,2,2]
=> ? = 1
[10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1,1,1,1]
=> ? = 4
[10,3,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> [4,3,2,1,1,1,1,1]
=> ? = 2
[10,2,1,1]
=> [4,2,1,1,1,1,1,1,1,1]
=> [8,2,2,1,1]
=> ? = 2
[9,5]
=> [2,2,2,2,2,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> ? = 5
[9,4,1]
=> [3,2,2,2,1,1,1,1,1]
=> [4,3,1,1,1,1,1,1,1]
=> ? = 3
[9,3,2]
=> [3,3,2,1,1,1,1,1,1]
=> [6,4,2,1,1]
=> ? = 1
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000149
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 69%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 69%
Values
[1]
=> [1]
=> [1]
=> [1]
=> 0
[2]
=> [1,1]
=> [2]
=> [1,1]
=> 0
[1,1]
=> [2]
=> [1,1]
=> [2]
=> 1
[3]
=> [1,1,1]
=> [2,1]
=> [2,1]
=> 0
[2,1]
=> [2,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,1]
=> [3]
=> [3]
=> [1,1,1]
=> 0
[4]
=> [1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 0
[3,1]
=> [2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[2,2]
=> [2,2]
=> [1,1,1,1]
=> [4]
=> 2
[2,1,1]
=> [3,1]
=> [3,1]
=> [2,1,1]
=> 0
[1,1,1,1]
=> [4]
=> [2,2]
=> [2,2]
=> 1
[5]
=> [1,1,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 0
[4,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[3,2]
=> [2,2,1]
=> [1,1,1,1,1]
=> [5]
=> 2
[3,1,1]
=> [3,1,1]
=> [3,2]
=> [2,2,1]
=> 0
[2,2,1]
=> [3,2]
=> [3,1,1]
=> [3,1,1]
=> 1
[2,1,1,1]
=> [4,1]
=> [2,2,1]
=> [3,2]
=> 1
[1,1,1,1,1]
=> [5]
=> [5]
=> [1,1,1,1,1]
=> 0
[6]
=> [1,1,1,1,1,1]
=> [4,2]
=> [2,2,1,1]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> [3,2,1]
=> [3,2,1]
=> 0
[3,3]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [6]
=> 3
[3,2,1]
=> [3,2,1]
=> [3,1,1,1]
=> [4,1,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [2,2,2]
=> [3,3]
=> 1
[2,2,2]
=> [3,3]
=> [6]
=> [1,1,1,1,1,1]
=> 0
[2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> [4,2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [5,1]
=> [2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> [6]
=> [3,3]
=> [2,2,2]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> [3,2,1,1]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> [4,3]
=> [2,2,2,1]
=> 0
[4,3]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> 3
[4,2,1]
=> [3,2,1,1]
=> [3,2,1,1]
=> [4,2,1]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> [2,2,2,1]
=> [4,3]
=> 1
[3,3,1]
=> [3,2,2]
=> [3,1,1,1,1]
=> [5,1,1]
=> 2
[3,2,2]
=> [3,3,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [2,2,1,1,1]
=> [5,2]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> 0
[2,2,2,1]
=> [4,3]
=> [3,2,2]
=> [3,3,1]
=> 1
[2,2,1,1,1]
=> [5,2]
=> [5,1,1]
=> [3,1,1,1,1]
=> 1
[2,1,1,1,1,1]
=> [6,1]
=> [3,3,1]
=> [3,2,2]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [4,2,1,1]
=> [4,2,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [4,1,1,1,1]
=> [5,1,1,1]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> [4,3,1]
=> [3,2,2,1]
=> 0
[5,3]
=> [2,2,2,1,1]
=> [2,1,1,1,1,1,1]
=> [7,1]
=> 3
[5,2,1]
=> [3,2,1,1,1]
=> [3,2,1,1,1]
=> [5,2,1]
=> 1
[4,4,2,1]
=> [4,3,2,2]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> ? = 3
[4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [7,2,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 1
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> ? = 1
[7,3,2]
=> [3,3,2,1,1,1,1]
=> [6,4,1,1]
=> [4,2,2,2,1,1]
=> ? = 1
[6,3,3]
=> [3,3,3,1,1,1]
=> [6,3,2,1]
=> [4,3,2,1,1,1]
=> ? = 0
[6,3,1,1,1]
=> [5,2,2,1,1,1]
=> [5,2,1,1,1,1,1]
=> [7,2,1,1,1]
=> ? = 2
[6,1,1,1,1,1,1]
=> [7,1,1,1,1,1]
=> [7,4,1]
=> [3,2,2,2,1,1,1]
=> ? = 0
[5,3,3,1]
=> [4,3,3,1,1]
=> [6,2,2,2]
=> [4,4,1,1,1,1]
=> ? = 1
[4,2,2,1,1,1,1]
=> [7,3,1,1]
=> [7,3,2]
=> [3,3,2,1,1,1,1]
=> ? = 0
[13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [8,4,1]
=> [3,2,2,2,1,1,1,1]
=> ? = 0
[10,2,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> [8,3,1,1]
=> [4,2,2,1,1,1,1,1]
=> ? = 1
[10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> [8,2,2,1]
=> [4,3,1,1,1,1,1,1]
=> ? = 1
[9,4]
=> [2,2,2,2,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> [10,1,1,1]
=> ? = 4
[9,2,2]
=> [3,3,1,1,1,1,1,1,1]
=> [6,4,2,1]
=> [4,3,2,2,1,1]
=> ? = 0
[9,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1]
=> [8,5]
=> [2,2,2,2,2,1,1,1]
=> ? = 0
[8,4,1]
=> [3,2,2,2,1,1,1,1]
=> [4,3,1,1,1,1,1,1]
=> [8,2,2,1]
=> ? = 3
[7,5,1]
=> [3,2,2,2,2,1,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> [10,2,1]
=> ? = 4
[6,6,1]
=> [3,2,2,2,2,2]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> [11,1,1]
=> ? = 5
[6,5,2]
=> [3,3,2,2,2,1]
=> [6,1,1,1,1,1,1,1]
=> [8,1,1,1,1,1]
=> ? = 3
[6,5,1,1]
=> [4,2,2,2,2,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [11,2]
=> ? = 5
[6,4,1,1,1]
=> [5,2,2,2,1,1]
=> [5,2,1,1,1,1,1,1]
=> [8,2,1,1,1]
=> ? = 3
[6,3,2,1,1]
=> [5,3,2,1,1,1]
=> [5,3,2,1,1,1]
=> [6,3,2,1,1]
=> ? = 1
[6,2,1,1,1,1,1]
=> [7,2,1,1,1,1]
=> [7,4,1,1]
=> [4,2,2,2,1,1,1]
=> ? = 1
[5,5,2,1]
=> [4,3,2,2,2]
=> [3,2,2,1,1,1,1,1,1]
=> [9,3,1]
=> ? = 4
[5,5,1,1,1]
=> [5,2,2,2,2]
=> [5,1,1,1,1,1,1,1,1]
=> [9,1,1,1,1]
=> ? = 4
[5,4,2,1,1]
=> [5,3,2,2,1]
=> [5,3,1,1,1,1,1]
=> [7,2,2,1,1]
=> ? = 2
[5,3,2,2,1]
=> [5,4,2,1,1]
=> [5,2,2,2,1,1]
=> [6,4,1,1,1]
=> ? = 2
[5,3,1,1,1,1,1]
=> [7,2,2,1,1]
=> [7,2,1,1,1,1]
=> [6,2,1,1,1,1,1]
=> ? = 2
[5,2,2,1,1,1,1]
=> [7,3,1,1,1]
=> [7,3,2,1]
=> [4,3,2,1,1,1,1]
=> ? = 0
[4,4,4,1]
=> [4,3,3,3]
=> [6,3,2,2]
=> [4,4,2,1,1,1]
=> ? = 1
[4,4,2,2,1]
=> [5,4,2,2]
=> [5,2,2,1,1,1,1]
=> [7,3,1,1,1]
=> ? = 3
[4,3,3,1,1,1]
=> [6,3,3,1]
=> [6,3,3,1]
=> [4,3,3,1,1,1]
=> ? = 1
[4,3,2,2,2]
=> [5,5,2,1]
=> [10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> ? = 1
[4,2,2,2,1,1,1]
=> [7,4,1,1]
=> [7,2,2,2]
=> [4,4,1,1,1,1,1]
=> ? = 1
[4,2,1,1,1,1,1,1,1]
=> [9,2,1,1]
=> [9,2,1,1]
=> [4,2,1,1,1,1,1,1,1]
=> ? = 1
[4,1,1,1,1,1,1,1,1,1]
=> [10,1,1,1]
=> [5,5,2,1]
=> [4,3,2,2,2]
=> ? = 1
[3,3,3,3,1]
=> [5,4,4]
=> [5,2,2,2,2]
=> [5,5,1,1,1]
=> ? = 2
[3,3,1,1,1,1,1,1,1]
=> [9,2,2]
=> [9,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1]
=> ? = 2
[3,2,1,1,1,1,1,1,1,1]
=> [10,2,1]
=> [5,5,1,1,1]
=> [5,2,2,2,2]
=> ? = 2
[2,2,2,2,2,2,1]
=> [7,6]
=> [7,3,3]
=> [3,3,3,1,1,1,1]
=> ? = 1
[14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 0
[13,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [8,4,1,1]
=> [4,2,2,2,1,1,1,1]
=> ? = 1
[12,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> [8,2,1,1,1,1]
=> [6,2,1,1,1,1,1,1]
=> ? = 2
[12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> [8,3,2,1]
=> [4,3,2,1,1,1,1,1]
=> ? = 0
[11,3]
=> [2,2,2,1,1,1,1,1,1,1,1]
=> [8,1,1,1,1,1,1]
=> [7,1,1,1,1,1,1,1]
=> ? = 3
[11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> [8,2,2,2]
=> [4,4,1,1,1,1,1,1]
=> ? = 1
[10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1,1,1,1]
=> [10,2,1,1]
=> ? = 4
[10,3,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> [4,3,2,1,1,1,1,1]
=> [8,3,2,1]
=> ? = 2
[10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> ? = 0
[9,3,1,1]
=> [4,2,2,1,1,1,1,1,1]
=> [4,2,2,2,1,1,1,1]
=> [8,4,1,1]
=> ? = 3
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000142
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 85%
St000142: Integer partitions ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 85%
Values
[1]
=> [1]
=> 0
[2]
=> [1,1]
=> 0
[1,1]
=> [2]
=> 1
[3]
=> [1,1,1]
=> 0
[2,1]
=> [2,1]
=> 1
[1,1,1]
=> [3]
=> 0
[4]
=> [1,1,1,1]
=> 0
[3,1]
=> [2,1,1]
=> 1
[2,2]
=> [2,2]
=> 2
[2,1,1]
=> [3,1]
=> 0
[1,1,1,1]
=> [4]
=> 1
[5]
=> [1,1,1,1,1]
=> 0
[4,1]
=> [2,1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> 2
[3,1,1]
=> [3,1,1]
=> 0
[2,2,1]
=> [3,2]
=> 1
[2,1,1,1]
=> [4,1]
=> 1
[1,1,1,1,1]
=> [5]
=> 0
[6]
=> [1,1,1,1,1,1]
=> 0
[5,1]
=> [2,1,1,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> 2
[4,1,1]
=> [3,1,1,1]
=> 0
[3,3]
=> [2,2,2]
=> 3
[3,2,1]
=> [3,2,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> 1
[2,2,2]
=> [3,3]
=> 0
[2,2,1,1]
=> [4,2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> 0
[1,1,1,1,1,1]
=> [6]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> 2
[5,1,1]
=> [3,1,1,1,1]
=> 0
[4,3]
=> [2,2,2,1]
=> 3
[4,2,1]
=> [3,2,1,1]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> 1
[3,3,1]
=> [3,2,2]
=> 2
[3,2,2]
=> [3,3,1]
=> 0
[3,2,1,1]
=> [4,2,1]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> 0
[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]
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> 2
[6,1,1]
=> [3,1,1,1,1,1]
=> 0
[5,3]
=> [2,2,2,1,1]
=> 3
[5,2,1]
=> [3,2,1,1,1]
=> 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> ? = 2
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> ? = 0
[8,2,1]
=> [3,2,1,1,1,1,1,1]
=> ? = 1
[8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> ? = 1
[7,4]
=> [2,2,2,2,1,1,1]
=> ? = 4
[7,3,1]
=> [3,2,2,1,1,1,1]
=> ? = 2
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 2
[7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> ? = 0
[6,4,1]
=> [3,2,2,2,1,1]
=> ? = 3
[6,3,2]
=> [3,3,2,1,1,1]
=> ? = 1
[6,3,1,1]
=> [4,2,2,1,1,1]
=> ? = 3
[6,2,2,1]
=> [4,3,1,1,1,1]
=> ? = 1
[5,2,1,1,1,1]
=> [6,2,1,1,1]
=> ? = 2
[5,1,1,1,1,1,1]
=> [7,1,1,1,1]
=> ? = 0
[4,3,1,1,1,1]
=> [6,2,2,1]
=> ? = 3
[4,2,2,1,1,1]
=> [6,3,1,1]
=> ? = 1
[4,2,1,1,1,1,1]
=> [7,2,1,1]
=> ? = 1
[4,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> ? = 1
[3,3,2,1,1,1]
=> [6,3,2]
=> ? = 2
[3,2,2,1,1,1,1]
=> [7,3,1]
=> ? = 0
[3,2,1,1,1,1,1,1]
=> [8,2,1]
=> ? = 2
[2,2,1,1,1,1,1,1,1]
=> [9,2]
=> ? = 1
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> ? = 0
[9,3]
=> [2,2,2,1,1,1,1,1,1]
=> ? = 3
[9,2,1]
=> [3,2,1,1,1,1,1,1,1]
=> ? = 1
[9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> ? = 1
[8,4]
=> [2,2,2,2,1,1,1,1]
=> ? = 4
[8,3,1]
=> [3,2,2,1,1,1,1,1]
=> ? = 2
[8,2,2]
=> [3,3,1,1,1,1,1,1]
=> ? = 0
[8,2,1,1]
=> [4,2,1,1,1,1,1,1]
=> ? = 2
[8,1,1,1,1]
=> [5,1,1,1,1,1,1,1]
=> ? = 0
[7,5]
=> [2,2,2,2,2,1,1]
=> ? = 5
[7,4,1]
=> [3,2,2,2,1,1,1]
=> ? = 3
[7,3,2]
=> [3,3,2,1,1,1,1]
=> ? = 1
[7,3,1,1]
=> [4,2,2,1,1,1,1]
=> ? = 3
[7,2,2,1]
=> [4,3,1,1,1,1,1]
=> ? = 1
[7,2,1,1,1]
=> [5,2,1,1,1,1,1]
=> ? = 1
[7,1,1,1,1,1]
=> [6,1,1,1,1,1,1]
=> ? = 1
[6,4,1,1]
=> [4,2,2,2,1,1]
=> ? = 4
[6,3,3]
=> [3,3,3,1,1,1]
=> ? = 0
[6,3,2,1]
=> [4,3,2,1,1,1]
=> ? = 2
[6,2,2,2]
=> [4,4,1,1,1,1]
=> ? = 2
[6,2,2,1,1]
=> [5,3,1,1,1,1]
=> ? = 0
[6,2,1,1,1,1]
=> [6,2,1,1,1,1]
=> ? = 2
[6,1,1,1,1,1,1]
=> [7,1,1,1,1,1]
=> ? = 0
[5,5,1,1]
=> [4,2,2,2,2]
=> ? = 5
[5,2,2,1,1,1]
=> [6,3,1,1,1]
=> ? = 1
[5,2,1,1,1,1,1]
=> [7,2,1,1,1]
=> ? = 1
[5,1,1,1,1,1,1,1]
=> [8,1,1,1,1]
=> ? = 1
Description
The number of even parts of a partition.
Matching statistic: St001232
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 54%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 54%
Values
[1]
=> [1,0]
=> [1,0]
=> 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 0
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 0
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 3
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 0
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 0
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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