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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St000870
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St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 2
[3]
=> 3
[2,1]
=> 3
[1,1,1]
=> 3
[4]
=> 4
[3,1]
=> 4
[2,2]
=> 3
[2,1,1]
=> 4
[1,1,1,1]
=> 4
[5]
=> 5
[4,1]
=> 5
[3,2]
=> 4
[3,1,1]
=> 5
[2,2,1]
=> 4
[2,1,1,1]
=> 5
[1,1,1,1,1]
=> 5
[6]
=> 6
[5,1]
=> 6
[4,2]
=> 5
[4,1,1]
=> 6
[3,3]
=> 8
[3,2,1]
=> 5
[3,1,1,1]
=> 6
[2,2,2]
=> 8
[2,2,1,1]
=> 5
[2,1,1,1,1]
=> 6
[1,1,1,1,1,1]
=> 6
[7]
=> 7
[6,1]
=> 7
[5,2]
=> 6
[5,1,1]
=> 7
[4,3]
=> 10
[4,2,1]
=> 6
[4,1,1,1]
=> 7
[3,3,1]
=> 10
[3,2,2]
=> 10
[3,2,1,1]
=> 6
[3,1,1,1,1]
=> 7
[2,2,2,1]
=> 10
[2,2,1,1,1]
=> 6
[2,1,1,1,1,1]
=> 7
[1,1,1,1,1,1,1]
=> 7
[8]
=> 8
[7,1]
=> 8
[6,2]
=> 7
[6,1,1]
=> 8
[5,3]
=> 12
[5,2,1]
=> 7
Description
The product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001880
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 21%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 21%
Values
[1]
=> [1,0]
=> [.,.]
=> ([],1)
=> ? = 1
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? = 2
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? = 2
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,2]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 4
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 8
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 8
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 6
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 10
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 6
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 10
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 10
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 6
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 10
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 6
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 7
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 12
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 7
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 15
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 12
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 12
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 7
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 15
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 12
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 12
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 7
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 15
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 12
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 7
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 8
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 14
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 8
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 18
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 14
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 21%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 21%
Values
[1]
=> [1,0]
=> [.,.]
=> ([],1)
=> ? = 1 - 1
[2]
=> [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? = 2 - 1
[1,1]
=> [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? = 2 - 1
[3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 3 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 4 - 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 5 - 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 8 - 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 5 - 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 8 - 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 6 - 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 10 - 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 6 - 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 10 - 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 10 - 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 6 - 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 10 - 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? = 6 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[.,.],.],.],.],.],.],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 7 - 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8 - 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8 - 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 7 - 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8 - 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 12 - 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 7 - 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8 - 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 15 - 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 12 - 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 12 - 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 7 - 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8 - 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 15 - 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 12 - 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 12 - 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [.,[[.,.],[[[[.,.],.],.],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 7 - 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[.,[[[[[[.,.],.],.],.],.],.]]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8 - 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 15 - 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? = 12 - 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 7 - 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8 - 1
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[[[[[[.,.],.],.],.],.],.],.],.]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8 - 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9 - 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9 - 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
=> ? = 8 - 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9 - 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 14 - 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
=> ? = 8 - 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9 - 1
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 18 - 1
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 14 - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
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