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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St001814
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001814: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 2
[2]
=> 3
[1,1]
=> 2
[3]
=> 4
[2,1]
=> 4
[1,1,1]
=> 2
[4]
=> 5
[3,1]
=> 6
[2,2]
=> 3
[2,1,1]
=> 4
[1,1,1,1]
=> 2
[5]
=> 6
[4,1]
=> 8
[3,2]
=> 6
[3,1,1]
=> 6
[2,2,1]
=> 4
[2,1,1,1]
=> 4
[1,1,1,1,1]
=> 2
[6]
=> 7
[5,1]
=> 10
[4,2]
=> 9
[4,1,1]
=> 8
[3,3]
=> 4
[3,2,1]
=> 8
[3,1,1,1]
=> 6
[2,2,2]
=> 3
[2,2,1,1]
=> 4
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 2
[7]
=> 8
[6,1]
=> 12
[5,2]
=> 12
[5,1,1]
=> 10
[4,3]
=> 8
[4,2,1]
=> 12
[4,1,1,1]
=> 8
[3,3,1]
=> 6
[3,2,2]
=> 6
[3,2,1,1]
=> 8
[3,1,1,1,1]
=> 6
[2,2,2,1]
=> 4
[2,2,1,1,1]
=> 4
[2,1,1,1,1,1]
=> 4
[1,1,1,1,1,1,1]
=> 2
[8]
=> 9
[7,1]
=> 14
[6,2]
=> 15
[6,1,1]
=> 12
[5,3]
=> 12
[5,2,1]
=> 16
Description
The number of partitions interlacing the given partition.
Matching statistic: St000708
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 2
[2]
=> 100 => [1,3] => [3,1]
=> 3
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 2
[3]
=> 1000 => [1,4] => [4,1]
=> 4
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 4
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 2
[4]
=> 10000 => [1,5] => [5,1]
=> 5
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 6
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 3
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 4
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 2
[5]
=> 100000 => [1,6] => [6,1]
=> 6
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 8
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 6
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 6
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 4
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 4
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 2
[6]
=> 1000000 => [1,7] => [7,1]
=> 7
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 10
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 9
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 8
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 4
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 8
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 6
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 3
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 4
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 2
[7]
=> 10000000 => [1,8] => [8,1]
=> 8
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 12
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 12
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 10
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 8
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 12
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 8
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 6
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 6
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 8
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 6
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 4
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 4
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 4
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 2
[8]
=> 100000000 => [1,9] => [9,1]
=> 9
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 14
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 15
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 12
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 12
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 16
Description
The product of the parts of an integer partition.
Matching statistic: St001232
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 50%
Values
[1]
=> [1,0]
=> [.,.]
=> [1,0]
=> 0 = 2 - 2
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> [1,1,0,0]
=> 0 = 2 - 2
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> [1,0,1,0]
=> 1 = 3 - 2
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0 = 2 - 2
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 4 - 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> ? = 4 - 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0 = 2 - 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 3 = 5 - 2
[2,2]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 2 = 4 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {3,6} - 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {3,6} - 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 2 - 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4 = 6 - 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 4 = 6 - 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {4,4,6,8} - 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {4,4,6,8} - 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,4,6,8} - 2
[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]
=> ? ∊ {4,4,6,8} - 2
[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 = 2 - 2
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 7 - 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 6 = 8 - 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {3,4,4,6,8,9,10} - 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 4 - 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {3,4,4,6,8,9,10} - 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {3,4,4,6,8,9,10} - 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {3,4,4,6,8,9,10} - 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {3,4,4,6,8,9,10} - 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {3,4,4,6,8,9,10} - 2
[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]
=> ? ∊ {3,4,4,6,8,9,10} - 2
[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 = 2 - 2
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6 = 8 - 2
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 8 = 10 - 2
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {4,4,4,6,6,8,8,8,12,12,12} - 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 6 - 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ? ∊ {4,4,4,6,6,8,8,8,12,12,12} - 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {4,4,4,6,6,8,8,8,12,12,12} - 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? ∊ {4,4,4,6,6,8,8,8,12,12,12} - 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {4,4,4,6,6,8,8,8,12,12,12} - 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ? ∊ {4,4,4,6,6,8,8,8,12,12,12} - 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {4,4,4,6,6,8,8,8,12,12,12} - 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {4,4,4,6,6,8,8,8,12,12,12} - 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,4,4,6,6,8,8,8,12,12,12} - 2
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {4,4,4,6,6,8,8,8,12,12,12} - 2
[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]
=> ? ∊ {4,4,4,6,6,8,8,8,12,12,12} - 2
[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]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 10 = 12 - 2
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 6 = 8 - 2
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 4 - 2
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[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]
=> ? ∊ {2,3,4,4,5,6,6,6,8,8,9,9,10,12,12,12,14,15,16} - 2
[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]
=> ? ∊ {2,4,4,4,4,4,6,8,8,8,8,8,10,10,12,12,12,12,12,12,14,16,16,16,18,18,20} - 2
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? ∊ {2,4,4,4,4,4,6,8,8,8,8,8,10,10,12,12,12,12,12,12,14,16,16,16,18,18,20} - 2
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? ∊ {2,4,4,4,4,4,6,8,8,8,8,8,10,10,12,12,12,12,12,12,14,16,16,16,18,18,20} - 2
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {2,4,4,4,4,4,6,8,8,8,8,8,10,10,12,12,12,12,12,12,14,16,16,16,18,18,20} - 2
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> 8 = 10 - 2
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {2,4,4,4,4,4,6,8,8,8,8,8,10,10,12,12,12,12,12,12,14,16,16,16,18,18,20} - 2
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? ∊ {2,4,4,4,4,4,6,8,8,8,8,8,10,10,12,12,12,12,12,12,14,16,16,16,18,18,20} - 2
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 4 = 6 - 2
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 6 - 2
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> 6 = 8 - 2
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 4 - 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001959
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001959: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 28%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001959: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 28%
Values
[1]
=> 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[2]
=> 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1]
=> 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[3]
=> 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[2,1]
=> 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,1,1]
=> 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[4]
=> 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[3,1]
=> 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 6
[2,2]
=> 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[2,1,1]
=> 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 4
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[5]
=> 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,4,6,6,8}
[4,1]
=> 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? ∊ {2,4,6,6,8}
[3,2]
=> 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 6
[3,1,1]
=> 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,4,6,6,8}
[2,2,1]
=> 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 4
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,6,6,8}
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,6,6,8}
[6]
=> 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {2,4,4,6,7,8,8,9,10}
[5,1]
=> 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {2,4,4,6,7,8,8,9,10}
[4,2]
=> 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? ∊ {2,4,4,6,7,8,8,9,10}
[4,1,1]
=> 1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,6,7,8,8,9,10}
[3,3]
=> 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[3,2,1]
=> 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,4,4,6,7,8,8,9,10}
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,6,7,8,8,9,10}
[2,2,2]
=> 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 3
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,6,7,8,8,9,10}
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,6,7,8,8,9,10}
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,6,7,8,8,9,10}
[7]
=> 10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[6,1]
=> 10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[5,2]
=> 1000100 => [1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[5,1,1]
=> 10000110 => [1,5,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[4,3]
=> 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[4,2,1]
=> 1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[3,3,1]
=> 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[3,2,2]
=> 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12}
[8]
=> 100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[7,1]
=> 100000010 => [1,7,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[6,2]
=> 10000100 => [1,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[6,1,1]
=> 100000110 => [1,6,1,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[5,3]
=> 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[5,2,1]
=> 10001010 => [1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[5,1,1,1]
=> 100001110 => [1,5,1,1,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[4,4]
=> 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[4,3,1]
=> 1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[4,2,2]
=> 1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[4,2,1,1]
=> 10010110 => [1,3,2,1,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[4,1,1,1,1]
=> 100011110 => [1,4,1,1,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[3,3,2]
=> 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[3,3,1,1]
=> 1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[3,2,2,1]
=> 1011010 => [1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[3,2,1,1,1]
=> 10101110 => [1,2,2,1,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[3,1,1,1,1,1]
=> 100111110 => [1,3,1,1,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[2,2,2,2]
=> 111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[2,2,2,1,1]
=> 1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[2,2,1,1,1,1]
=> 11011110 => [1,1,2,1,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
[2,1,1,1,1,1,1]
=> 101111110 => [1,2,1,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3,4,4,4,5,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16}
Description
The product of the heights of the peaks of a Dyck path.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 28%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 28%
Values
[1]
=> 10 => [1,2] => ([(1,2)],3)
=> 1 = 2 - 1
[2]
=> 100 => [1,3] => ([(2,3)],4)
=> 1 = 2 - 1
[1,1]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[3]
=> 1000 => [1,4] => ([(3,4)],5)
=> 1 = 2 - 1
[2,1]
=> 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,1,1]
=> 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 4 - 1
[4]
=> 10000 => [1,5] => ([(4,5)],6)
=> 1 = 2 - 1
[3,1]
=> 10010 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6} - 1
[2,2]
=> 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 3 - 1
[2,1,1]
=> 10110 => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,6} - 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[5]
=> 100000 => [1,6] => ([(5,6)],7)
=> 1 = 2 - 1
[4,1]
=> 100010 => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,4,6,6,8} - 1
[3,2]
=> 10100 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,6,6,8} - 1
[3,1,1]
=> 100110 => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,4,6,6,8} - 1
[2,2,1]
=> 11010 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {4,4,6,6,8} - 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {4,4,6,6,8} - 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 6 - 1
[6]
=> 1000000 => [1,7] => ([(6,7)],8)
=> ? ∊ {2,4,4,6,7,8,8,9,10} - 1
[5,1]
=> 1000010 => [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,6,7,8,8,9,10} - 1
[4,2]
=> 100100 => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,4,4,6,7,8,8,9,10} - 1
[4,1,1]
=> 1000110 => [1,4,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,6,7,8,8,9,10} - 1
[3,3]
=> 11000 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 2 = 3 - 1
[3,2,1]
=> 101010 => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,4,4,6,7,8,8,9,10} - 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => ([(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,6,7,8,8,9,10} - 1
[2,2,2]
=> 11100 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 4 - 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,4,4,6,7,8,8,9,10} - 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,6,7,8,8,9,10} - 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,6,7,8,8,9,10} - 1
[7]
=> 10000000 => [1,8] => ([(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[6,1]
=> 10000010 => [1,6,2] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[5,2]
=> 1000100 => [1,4,3] => ([(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[5,1,1]
=> 10000110 => [1,5,1,2] => ([(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[4,3]
=> 101000 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[4,2,1]
=> 1001010 => [1,3,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => ([(1,6),(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[3,3,1]
=> 110010 => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[3,2,2]
=> 101100 => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => ([(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => ([(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,8,10,12,12,12} - 1
[8]
=> 100000000 => [1,9] => ([(8,9)],10)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[7,1]
=> 100000010 => [1,7,2] => ([(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[6,2]
=> 10000100 => [1,5,3] => ([(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[6,1,1]
=> 100000110 => [1,6,1,2] => ([(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[5,3]
=> 1001000 => [1,3,4] => ([(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[5,2,1]
=> 10001010 => [1,4,2,2] => ([(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[5,1,1,1]
=> 100001110 => [1,5,1,1,2] => ([(1,7),(1,8),(1,9),(2,7),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[4,4]
=> 110000 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 2 = 3 - 1
[4,3,1]
=> 1010010 => [1,2,3,2] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[4,2,2]
=> 1001100 => [1,3,1,3] => ([(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[4,2,1,1]
=> 10010110 => [1,3,2,1,2] => ([(1,7),(1,8),(2,6),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[4,1,1,1,1]
=> 100011110 => [1,4,1,1,1,2] => ([(1,6),(1,7),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[3,3,2]
=> 110100 => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[3,3,1,1]
=> 1100110 => [1,1,3,1,2] => ([(1,6),(1,7),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[3,2,2,1]
=> 1011010 => [1,2,1,2,2] => ([(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[3,2,1,1,1]
=> 10101110 => [1,2,2,1,1,2] => ([(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[3,1,1,1,1,1]
=> 100111110 => [1,3,1,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[2,2,2,2]
=> 111100 => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 5 - 1
[2,2,2,1,1]
=> 1110110 => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[2,2,1,1,1,1]
=> 11011110 => [1,1,2,1,1,1,2] => ([(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {2,4,4,4,6,6,6,8,8,8,9,9,10,12,12,12,12,14,15,16} - 1
[3,3,3]
=> 111000 => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 4 - 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
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