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Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000548
St000548: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 3
[1,1,1]
=> 3
[4]
=> 1
[3,1]
=> 3
[2,2]
=> 2
[2,1,1]
=> 4
[1,1,1,1]
=> 4
[5]
=> 1
[4,1]
=> 3
[3,2]
=> 3
[3,1,1]
=> 5
[2,2,1]
=> 5
[2,1,1,1]
=> 5
[1,1,1,1,1]
=> 5
[6]
=> 1
[5,1]
=> 3
[4,2]
=> 3
[4,1,1]
=> 5
[3,3]
=> 2
[3,2,1]
=> 6
[3,1,1,1]
=> 6
[2,2,2]
=> 3
[2,2,1,1]
=> 6
[2,1,1,1,1]
=> 6
[1,1,1,1,1,1]
=> 6
[7]
=> 1
[6,1]
=> 3
[5,2]
=> 3
[5,1,1]
=> 5
[4,3]
=> 3
[4,2,1]
=> 7
[4,1,1,1]
=> 7
[3,3,1]
=> 5
[3,2,2]
=> 5
[3,2,1,1]
=> 7
[3,1,1,1,1]
=> 7
[2,2,2,1]
=> 7
[2,2,1,1,1]
=> 7
[2,1,1,1,1,1]
=> 7
[1,1,1,1,1,1,1]
=> 7
[8]
=> 1
[7,1]
=> 3
[6,2]
=> 3
[6,1,1]
=> 5
[5,3]
=> 3
[5,2,1]
=> 7
Description
The number of different non-empty partial sums of an integer partition.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 50%
Values
[1]
=> 10 => [1,1] => [1,0,1,0]
=> 1
[2]
=> 100 => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1]
=> 110 => [2,1] => [1,1,0,0,1,0]
=> 1
[3]
=> 1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[2,1]
=> 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ? = 3
[1,1,1]
=> 1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4]
=> 10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[3,1]
=> 10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {3,4}
[2,2]
=> 1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,1,1]
=> 10110 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {3,4}
[1,1,1,1]
=> 11110 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[5]
=> 100000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[4,1]
=> 100010 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {3,3,5,5}
[3,2]
=> 10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {3,3,5,5}
[3,1,1]
=> 100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[2,2,1]
=> 11010 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {3,3,5,5}
[2,1,1,1]
=> 101110 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {3,3,5,5}
[1,1,1,1,1]
=> 111110 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[6]
=> 1000000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,1]
=> 1000010 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {3,3,5,6,6}
[4,2]
=> 100100 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {3,3,5,6,6}
[4,1,1]
=> 1000110 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
[3,3]
=> 11000 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[3,2,1]
=> 101010 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,5,6,6}
[3,1,1,1]
=> 1001110 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[2,2,2]
=> 11100 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[2,2,1,1]
=> 110110 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {3,3,5,6,6}
[2,1,1,1,1]
=> 1011110 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {3,3,5,6,6}
[1,1,1,1,1,1]
=> 1111110 => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[7]
=> 10000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[6,1]
=> 10000010 => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[5,2]
=> 1000100 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[5,1,1]
=> 10000110 => [1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[4,3]
=> 101000 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[4,2,1]
=> 1001010 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[4,1,1,1]
=> 10001110 => [1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[3,3,1]
=> 110010 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[3,2,2]
=> 101100 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[3,2,1,1]
=> 1010110 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[3,1,1,1,1]
=> 10011110 => [1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[2,2,2,1]
=> 111010 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[2,2,1,1,1]
=> 1101110 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[2,1,1,1,1,1]
=> 10111110 => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[1,1,1,1,1,1,1]
=> 11111110 => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,3,3,3,5,5,5,7,7,7,7,7,7,7,7}
[8]
=> 100000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[7,1]
=> 100000010 => [1,6,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[6,2]
=> 10000100 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[6,1,1]
=> 100000110 => [1,5,2,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[5,3]
=> 1001000 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[5,2,1]
=> 10001010 => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[5,1,1,1]
=> 100001110 => [1,4,3,1] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[4,4]
=> 110000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 4
[4,3,1]
=> 1010010 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[4,2,2]
=> 1001100 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[4,2,1,1]
=> 10010110 => [1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[4,1,1,1,1]
=> 100011110 => [1,3,4,1] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[3,3,2]
=> 110100 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[3,3,1,1]
=> 1100110 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[3,2,2,1]
=> 1011010 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[3,2,1,1,1]
=> 10101110 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[3,1,1,1,1,1]
=> 100111110 => [1,2,5,1] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[2,2,2,2]
=> 111100 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[2,2,2,1,1]
=> 1110110 => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[2,2,1,1,1,1]
=> 11011110 => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[2,1,1,1,1,1,1]
=> 101111110 => [1,1,6,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[1,1,1,1,1,1,1,1]
=> 111111110 => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {1,3,3,3,4,5,7,7,8,8,8,8,8,8,8,8,8,8}
[9]
=> 1000000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,3,3,3,3,5,5,5,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9}
[8,1]
=> 1000000010 => [1,7,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {1,3,3,3,3,5,5,5,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9}
[7,2]
=> 100000100 => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,3,3,3,3,5,5,5,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9}
[7,1,1]
=> 1000000110 => [1,6,2,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {1,3,3,3,3,5,5,5,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9}
[6,3]
=> 10001000 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,3,3,3,3,5,5,5,7,7,7,7,7,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9}
[3,3,3]
=> 111000 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[5,5]
=> 1100000 => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[2,2,2,2,2]
=> 1111100 => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2
[4,4,4]
=> 1110000 => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 4
[3,3,3,3]
=> 1111000 => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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