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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St000566
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(load all 2 compositions to match this statistic)
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 1
[.,[.,.]]
=> [[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 3
[[.,.],.]
=> [[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 3
[.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [4,1,1,1]
=> 6
[.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [4,1,1,1]
=> 6
[[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7)
=> [3,2,1,1]
=> 4
[[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [4,1,1,1]
=> 6
[[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7)
=> [4,1,1,1]
=> 6
[.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> [5,1,1,1,1]
=> 10
[.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> [5,1,1,1,1]
=> 10
[.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9)
=> [4,2,1,1,1]
=> 7
[.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> [5,1,1,1,1]
=> 10
[.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> [5,1,1,1,1]
=> 10
[[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 7
[[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 7
[[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 7
[[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 7
[[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> [5,1,1,1,1]
=> 10
[[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> [5,1,1,1,1]
=> 10
[[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9)
=> [4,2,1,1,1]
=> 7
[[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> [5,1,1,1,1]
=> 10
[[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> [5,1,1,1,1]
=> 10
[.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> 15
[.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> 15
[.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> [5,2,1,1,1,1]
=> 11
[.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> 15
[.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> 15
[.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> 11
[.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> 11
[.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> 11
[.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> [5,2,1,1,1,1]
=> 11
[.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> 15
[.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> 15
[.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> [5,2,1,1,1,1]
=> 11
[.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> 15
[.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> [6,1,1,1,1,1]
=> 15
[[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> 11
[[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> 11
[[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> [4,2,2,1,1,1]
=> 8
[[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> 11
[[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> 11
[[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> 9
[[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> 9
[[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> 9
[[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> [4,3,1,1,1,1]
=> 9
[[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> 11
[[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> 11
[[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> [4,2,2,1,1,1]
=> 8
[[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> 11
[[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> [5,2,1,1,1,1]
=> 11
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000569
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St000569: Binary trees ⟶ ℤResult quality: 90% ●values known / values provided: 98%●distinct values known / distinct values provided: 90%
Values
[.,.]
=> ? = 1
[.,[.,.]]
=> 3
[[.,.],.]
=> 3
[.,[.,[.,.]]]
=> 6
[.,[[.,.],.]]
=> 6
[[.,.],[.,.]]
=> 4
[[.,[.,.]],.]
=> 6
[[[.,.],.],.]
=> 6
[.,[.,[.,[.,.]]]]
=> 10
[.,[.,[[.,.],.]]]
=> 10
[.,[[.,.],[.,.]]]
=> 7
[.,[[.,[.,.]],.]]
=> 10
[.,[[[.,.],.],.]]
=> 10
[[.,.],[.,[.,.]]]
=> 7
[[.,.],[[.,.],.]]
=> 7
[[.,[.,.]],[.,.]]
=> 7
[[[.,.],.],[.,.]]
=> 7
[[.,[.,[.,.]]],.]
=> 10
[[.,[[.,.],.]],.]
=> 10
[[[.,.],[.,.]],.]
=> 7
[[[.,[.,.]],.],.]
=> 10
[[[[.,.],.],.],.]
=> 10
[.,[.,[.,[.,[.,.]]]]]
=> 15
[.,[.,[.,[[.,.],.]]]]
=> 15
[.,[.,[[.,.],[.,.]]]]
=> 11
[.,[.,[[.,[.,.]],.]]]
=> 15
[.,[.,[[[.,.],.],.]]]
=> 15
[.,[[.,.],[.,[.,.]]]]
=> 11
[.,[[.,.],[[.,.],.]]]
=> 11
[.,[[.,[.,.]],[.,.]]]
=> 11
[.,[[[.,.],.],[.,.]]]
=> 11
[.,[[.,[.,[.,.]]],.]]
=> 15
[.,[[.,[[.,.],.]],.]]
=> 15
[.,[[[.,.],[.,.]],.]]
=> 11
[.,[[[.,[.,.]],.],.]]
=> 15
[.,[[[[.,.],.],.],.]]
=> 15
[[.,.],[.,[.,[.,.]]]]
=> 11
[[.,.],[.,[[.,.],.]]]
=> 11
[[.,.],[[.,.],[.,.]]]
=> 8
[[.,.],[[.,[.,.]],.]]
=> 11
[[.,.],[[[.,.],.],.]]
=> 11
[[.,[.,.]],[.,[.,.]]]
=> 9
[[.,[.,.]],[[.,.],.]]
=> 9
[[[.,.],.],[.,[.,.]]]
=> 9
[[[.,.],.],[[.,.],.]]
=> 9
[[.,[.,[.,.]]],[.,.]]
=> 11
[[.,[[.,.],.]],[.,.]]
=> 11
[[[.,.],[.,.]],[.,.]]
=> 8
[[[.,[.,.]],.],[.,.]]
=> 11
[[[[.,.],.],.],[.,.]]
=> 11
[[.,[.,[.,[.,.]]]],.]
=> 15
Description
The sum of the heights of the vertices of a binary tree.
The height of a vertex is the number of edges to a leaf, see [3].
Matching statistic: St000448
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000448: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 50%
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000448: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0 = 1 - 1
[.,[.,.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[[.,.],.]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5 = 6 - 1
[.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5 = 6 - 1
[[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 4 - 1
[[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5 = 6 - 1
[[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5 = 6 - 1
[.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 6 = 7 - 1
[.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6 = 7 - 1
[[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6 = 7 - 1
[[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6 = 7 - 1
[[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6 = 7 - 1
[[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 6 = 7 - 1
[[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> ([(0,9),(1,9),(2,8),(3,8),(4,7),(5,7),(6,9),(6,10),(7,10),(8,10)],11)
=> ([(0,6),(1,5),(2,4),(3,4),(3,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(6,7),(6,8)],9)
=> ? = 9 - 1
[[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(6,7),(6,8)],9)
=> ? = 9 - 1
[[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(6,7),(6,8)],9)
=> ? = 9 - 1
[[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(6,7),(6,8)],9)
=> ? = 9 - 1
[[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> ([(0,9),(1,9),(2,8),(3,8),(4,7),(5,7),(6,9),(6,10),(7,10),(8,10)],11)
=> ([(0,6),(1,5),(2,4),(3,4),(3,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,[.,[.,[.,.]]]],.]
=> [[[],[[],[[],[[],[]]]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[.,[.,[[.,.],.]]],.]
=> [[[],[[],[[[],[]],[]]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[.,[[.,[.,.]],.]],.]
=> [[[],[[[],[[],[]]],[]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[.,[[[.,.],.],.]],.]
=> [[[],[[[[],[]],[]],[]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[[.,.],[.,[.,.]]],.]
=> [[[[],[]],[[],[[],[]]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[[[.,.],.],[.,.]],.]
=> [[[[[],[]],[]],[[],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[[.,[.,[.,.]]],.],.]
=> [[[[],[[],[[],[]]]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[[.,[[.,.],.]],.],.]
=> [[[[],[[[],[]],[]]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[[[.,.],[.,.]],.],.]
=> [[[[[],[]],[[],[]]],[]],[]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[[[.,[.,.]],.],.],.]
=> [[[[[],[[],[]]],[]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[[[[.,.],.],.],.],.]
=> [[[[[[],[]],[]],[]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
Description
The number of pairs of vertices of a graph with distance 2.
This is the coefficient of the quadratic term of the Wiener polynomial.
Matching statistic: St001308
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001308: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 50%
Mp00046: Ordered trees —to graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001308: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 50%
Values
[.,.]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0 = 1 - 1
[.,[.,.]]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[[.,.],.]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 3 - 1
[.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5 = 6 - 1
[.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5 = 6 - 1
[[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 4 - 1
[[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5 = 6 - 1
[[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5 = 6 - 1
[.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 6 = 7 - 1
[.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6 = 7 - 1
[[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6 = 7 - 1
[[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6 = 7 - 1
[[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 6 = 7 - 1
[[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 6 = 7 - 1
[[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> ([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(1,6),(2,4),(3,5),(4,6),(5,7),(6,7)],8)
=> ? = 10 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> ([(0,9),(1,9),(2,8),(3,8),(4,7),(5,7),(6,9),(6,10),(7,10),(8,10)],11)
=> ([(0,6),(1,5),(2,4),(3,4),(3,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(6,7),(6,8)],9)
=> ? = 9 - 1
[[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(6,7),(6,8)],9)
=> ? = 9 - 1
[[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(6,7),(6,8)],9)
=> ? = 9 - 1
[[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> ([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)
=> ([(0,8),(1,7),(2,4),(3,5),(4,7),(5,8),(6,7),(6,8)],9)
=> ? = 9 - 1
[[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> ([(0,9),(1,9),(2,8),(3,8),(4,7),(5,7),(6,9),(6,10),(7,10),(8,10)],11)
=> ([(0,6),(1,5),(2,4),(3,4),(3,7),(5,7),(6,7)],8)
=> ? = 8 - 1
[[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> ([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)
=> ([(0,8),(1,7),(2,6),(3,4),(4,7),(5,6),(5,8),(7,8)],9)
=> ? = 11 - 1
[[.,[.,[.,[.,.]]]],.]
=> [[[],[[],[[],[[],[]]]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[.,[.,[[.,.],.]]],.]
=> [[[],[[],[[[],[]],[]]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[.,[[.,[.,.]],.]],.]
=> [[[],[[[],[[],[]]],[]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[.,[[[.,.],.],.]],.]
=> [[[],[[[[],[]],[]],[]]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[[.,.],[.,[.,.]]],.]
=> [[[[],[]],[[],[[],[]]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[[[.,.],.],[.,.]],.]
=> [[[[[],[]],[]],[[],[]]],[]]
=> ([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[[.,[.,[.,.]]],.],.]
=> [[[[],[[],[[],[]]]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[[.,[[.,.],.]],.],.]
=> [[[[],[[[],[]],[]]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[[[.,.],[.,.]],.],.]
=> [[[[[],[]],[[],[]]],[]],[]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,7),(1,5),(2,4),(3,6),(4,8),(5,8),(6,7),(7,8)],9)
=> ? = 11 - 1
[[[[.,[.,.]],.],.],.]
=> [[[[[],[[],[]]],[]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
[[[[[.,.],.],.],.],.]
=> [[[[[[],[]],[]],[]],[]],[]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)
=> ([(0,7),(1,9),(2,8),(3,5),(4,6),(5,8),(6,9),(7,8),(7,9)],10)
=> ? = 15 - 1
Description
The number of induced paths on three vertices in a graph.
Matching statistic: St000219
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000219: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 30%
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000219: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 30%
Values
[.,.]
=> [[],[]]
=> [1,0,1,0]
=> [1,2] => ? = 1 - 2
[.,[.,.]]
=> [[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 3 - 2
[[.,.],.]
=> [[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1 = 3 - 2
[.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,3,5,6,2,4] => 4 = 6 - 2
[.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,4,5,2,6,3] => 4 = 6 - 2
[[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,3,1,5,6,4] => 2 = 4 - 2
[[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,4,5,1,3,6] => 4 = 6 - 2
[[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [3,4,1,5,2,6] => 4 = 6 - 2
[.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,5,7,8,2,4,6] => ? = 10 - 2
[.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,3,6,7,2,8,4,5] => ? = 10 - 2
[.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,4,5,2,7,8,3,6] => ? = 7 - 2
[.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,4,6,7,2,3,8,5] => ? = 10 - 2
[.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,5,6,2,7,3,8,4] => ? = 10 - 2
[[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,3,1,5,7,8,4,6] => ? = 7 - 2
[[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,3,1,6,7,4,8,5] => ? = 7 - 2
[[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [2,4,5,1,3,7,8,6] => ? = 7 - 2
[[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2,7,8,6] => ? = 7 - 2
[[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [2,4,6,7,1,3,5,8] => ? = 10 - 2
[[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [2,5,6,1,7,3,4,8] => ? = 10 - 2
[[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [3,4,1,6,7,2,5,8] => ? = 7 - 2
[[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [3,5,6,1,2,7,4,8] => ? = 10 - 2
[[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [4,5,1,6,2,7,3,8] => ? = 10 - 2
[.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,3,5,7,9,10,2,4,6,8] => ? = 15 - 2
[.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,3,5,8,9,2,10,4,6,7] => ? = 15 - 2
[.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,3,6,7,2,9,10,4,5,8] => ? = 11 - 2
[.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,3,6,8,9,2,4,10,5,7] => ? = 15 - 2
[.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,3,7,8,2,9,4,10,5,6] => ? = 15 - 2
[.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,5,2,7,9,10,3,6,8] => ? = 11 - 2
[.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,4,5,2,8,9,3,10,6,7] => ? = 11 - 2
[.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,4,6,7,2,3,9,10,5,8] => ? = 11 - 2
[.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,5,6,2,7,3,9,10,4,8] => ? = 11 - 2
[.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,4,6,8,9,2,3,5,10,7] => ? = 15 - 2
[.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,4,7,8,2,9,3,5,10,6] => ? = 15 - 2
[.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,2,8,9,3,4,10,7] => ? = 11 - 2
[.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,5,7,8,2,3,9,4,10,6] => ? = 15 - 2
[.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,7,2,8,3,9,4,10,5] => ? = 15 - 2
[[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [2,3,1,5,7,9,10,4,6,8] => ? = 11 - 2
[[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [2,3,1,5,8,9,4,10,6,7] => ? = 11 - 2
[[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [2,3,1,6,7,4,9,10,5,8] => ? = 8 - 2
[[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [2,3,1,6,8,9,4,5,10,7] => ? = 11 - 2
[[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [2,3,1,7,8,4,9,5,10,6] => ? = 11 - 2
[[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3,7,9,10,6,8] => ? = 9 - 2
[[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,5,1,3,8,9,6,10,7] => ? = 9 - 2
[[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [3,4,1,5,2,7,9,10,6,8] => ? = 9 - 2
[[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2,8,9,6,10,7] => ? = 9 - 2
[[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [2,4,6,7,1,3,5,9,10,8] => ? = 11 - 2
[[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [2,5,6,1,7,3,4,9,10,8] => ? = 11 - 2
[[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [3,4,1,6,7,2,5,9,10,8] => ? = 8 - 2
[[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [3,5,6,1,2,7,4,9,10,8] => ? = 11 - 2
[[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,7,3,9,10,8] => ? = 11 - 2
[[.,[.,[.,[.,.]]]],.]
=> [[[],[[],[[],[[],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [2,4,6,8,9,1,3,5,7,10] => ? = 15 - 2
[[.,[.,[[.,.],.]]],.]
=> [[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [2,4,7,8,1,9,3,5,6,10] => ? = 15 - 2
[[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [2,5,6,1,8,9,3,4,7,10] => ? = 11 - 2
[[.,[[.,[.,.]],.]],.]
=> [[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [2,5,7,8,1,3,9,4,6,10] => ? = 15 - 2
[[.,[[[.,.],.],.]],.]
=> [[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [2,6,7,1,8,3,9,4,5,10] => ? = 15 - 2
[[[.,.],[.,[.,.]]],.]
=> [[[[],[]],[[],[[],[]]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,4,1,6,8,9,2,5,7,10] => ? = 11 - 2
[[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [3,4,1,7,8,2,9,5,6,10] => ? = 11 - 2
Description
The number of occurrences of the pattern 231 in a permutation.
Matching statistic: St000264
Values
[.,.]
=> ([],1)
=> ([],1)
=> ? = 1 - 5
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 3 - 5
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ? = 3 - 5
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 6 - 5
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 6 - 5
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 4 - 5
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 6 - 5
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 6 - 5
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 10 - 5
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 10 - 5
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ? = 7 - 5
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 10 - 5
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 10 - 5
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 7 - 5
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 7 - 5
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 7 - 5
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 7 - 5
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 10 - 5
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 10 - 5
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ? = 7 - 5
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 10 - 5
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 10 - 5
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ? = 11 - 5
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? = 11 - 5
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? = 11 - 5
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? = 11 - 5
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? = 11 - 5
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ? = 11 - 5
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 11 - 5
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 11 - 5
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 8 - 5
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 11 - 5
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 11 - 5
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 9 - 5
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 9 - 5
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 9 - 5
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 9 - 5
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 11 - 5
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 11 - 5
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 8 - 5
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 11 - 5
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ? = 11 - 5
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ? = 11 - 5
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ? = 15 - 5
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ? = 11 - 5
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000668
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[.,.]
=> ([],1)
=> [1]
=> []
=> ? = 1 - 7
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 3 - 7
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 3 - 7
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> ? = 4 - 7
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 8 - 7
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 8 - 7
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
Description
The least common multiple of the parts of the partition.
Matching statistic: St000707
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[.,.]
=> ([],1)
=> [1]
=> []
=> ? = 1 - 7
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 3 - 7
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 3 - 7
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> ? = 4 - 7
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 8 - 7
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 8 - 7
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
Description
The product of the factorials of the parts.
Matching statistic: St000708
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[.,.]
=> ([],1)
=> [1]
=> []
=> ? = 1 - 7
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 3 - 7
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 3 - 7
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> ? = 4 - 7
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 8 - 7
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 8 - 7
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
Description
The product of the parts of an integer partition.
Matching statistic: St000770
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000770: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[.,.]
=> ([],1)
=> [1]
=> []
=> ? = 1 - 7
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 3 - 7
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> []
=> ? = 3 - 7
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1]
=> ? = 4 - 7
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> []
=> ? = 6 - 7
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> ? = 7 - 7
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> []
=> ? = 10 - 7
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 8 - 7
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2 = 9 - 7
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1 = 8 - 7
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
[[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> []
=> ? = 15 - 7
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> ? = 11 - 7
Description
The major index of an integer partition when read from bottom to top.
This is the sum of the positions of the corners of the shape of an integer partition when reading from bottom to top.
For example, the partition $\lambda = (8,6,6,4,3,3)$ has corners at positions 3,6,9, and 13, giving a major index of 31.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000478Another weight of a partition according to Alladi. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition.
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