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Your data matches 70 different statistics following compositions of up to 3 maps.
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Matching statistic: St001714
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: 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
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1]
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> 0
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> 0
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> 0
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> 0
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> 0
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> 0
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1]
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1]
=> 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [5,1]
=> [1]
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [5,1]
=> [1]
=> 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [5,1]
=> [1]
=> 0
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [5,1]
=> [1]
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [5,1]
=> [1]
=> 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> 0
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> 0
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [4,1,1]
=> [1,1]
=> 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> 0
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> 0
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> 0
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2]
=> 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> 0
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [5,1]
=> [1]
=> 0
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition.
In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St000455
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 0
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 0
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> 0
[.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> 0
[[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0
[[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0
[[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0
[[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 0
[[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 0
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[[.,.],[[.,.],[.,.]]],.]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[.,[[[[.,.],[.,.]],[.,.]],.]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[.,.],[.,[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[.,.],[.,[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000454
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> 1 = 0 + 1
[[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[[.,.],[[.,.],[.,.]]],.]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[.,[[[[.,.],[.,.]],[.,.]],.]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[.,[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[.,[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 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.
Matching statistic: St000422
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[.,.],[[.,.],[.,.]]],.]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[[.,.],[.,.]],[.,.]],.]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St001330
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> ([(1,2)],3)
=> 2 = 0 + 2
[[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 2
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[.,.],[.,[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[.,.],[[.,.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[.,[.,.]],[.,.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[[.,.],.],[.,.]],[.,.]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[[.,.],[.,.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[.,.],[[.,.],[.,.]]],.]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[.,[[[[.,.],[.,.]],[.,.]],.]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7)
=> ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[.,[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7)
=> ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,.],[.,[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
[[.,.],[[.,.],[.,[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001934
Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 51%●distinct values known / distinct values provided: 25%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 51%●distinct values known / distinct values provided: 25%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> [2,2,1,1]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 1 = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 1 = 0 + 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [4,2]
=> [2,2,1,1]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 1 + 1
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 1 + 1
[[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 0 + 1
[[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 0 + 1
[[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 0 + 1
[[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 0 + 1
[[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 1 + 1
[[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[[[[.,.],.],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[[[[.,.],[.,.]],.],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 1 + 1
[[[.,.],[[.,.],[.,.]]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[[[[.,.],[.,.]],[.,.]],.]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 1 + 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 1 + 1
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> [5,5]
=> [2,2,2,2,2]
=> ? = 0 + 1
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$
(a_1, b_1),\dots,(a_r, b_r)
$$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Matching statistic: St001940
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Values
[[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7)
=> [3,2,1,1]
=> 0
[.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9)
=> [4,2,1,1,1]
=> 0
[[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 0
[[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 0
[[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 0
[[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 0
[[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9)
=> [4,2,1,1,1]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> ([(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]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> ([(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]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> ([(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]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> ([(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]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> ([(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]
=> 0
[.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> ([(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]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ([(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]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> ([(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]
=> 0
[[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> ([(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]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> ([(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]
=> 0
[[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> ([(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]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ([(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]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> ([(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]
=> 0
[[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> ([(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]
=> 0
[[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> ([(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]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> ([(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]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> ([(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]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> ([(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]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> ([(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]
=> 0
[[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> ([(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]
=> 0
[[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> ([(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]
=> 0
[[[.,.],[.,[.,.]]],.]
=> [[[[],[]],[[],[[],[]]]],[]]
=> ([(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]
=> 0
[[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> ([(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]
=> 0
[[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> ([(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]
=> 0
[[[[.,.],.],[.,.]],.]
=> [[[[[],[]],[]],[[],[]]],[]]
=> ([(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]
=> 0
[[[[.,.],[.,.]],.],.]
=> [[[[[],[]],[[],[]]],[]],[]]
=> ([(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]
=> 0
[.,[.,[.,[[.,.],[.,.]]]]]
=> [[],[[],[[],[[[],[]],[[],[]]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[.,[[.,.],[.,[.,.]]]]]
=> [[],[[],[[[],[]],[[],[[],[]]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[.,[[.,.],[[.,.],.]]]]
=> [[],[[],[[[],[]],[[[],[]],[]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[.,[[.,[.,.]],[.,.]]]]
=> [[],[[],[[[],[[],[]]],[[],[]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[.,[[[.,.],.],[.,.]]]]
=> [[],[[],[[[[],[]],[]],[[],[]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[.,[[[.,.],[.,.]],.]]]
=> [[],[[],[[[[],[]],[[],[]]],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[[.,.],[.,[.,[.,.]]]]]
=> [[],[[[],[]],[[],[[],[[],[]]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[[.,.],[.,[[.,.],.]]]]
=> [[],[[[],[]],[[],[[[],[]],[]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[[.,.],[[.,.],[.,.]]]]
=> [[],[[[],[]],[[[],[]],[[],[]]]]]
=> ([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> [5,2,2,1,1,1,1]
=> 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [[],[[[],[]],[[[],[[],[]]],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[[.,.],[[[.,.],.],.]]]
=> [[],[[[],[]],[[[[],[]],[]],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[[.,[.,.]],[.,[.,.]]]]
=> [[],[[[],[[],[]]],[[],[[],[]]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> 0
[.,[[.,[.,.]],[[.,.],.]]]
=> [[],[[[],[[],[]]],[[[],[]],[]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> 0
[.,[[[.,.],.],[.,[.,.]]]]
=> [[],[[[[],[]],[]],[[],[[],[]]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> 0
[.,[[[.,.],.],[[.,.],.]]]
=> [[],[[[[],[]],[]],[[[],[]],[]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> 0
[.,[[.,[.,[.,.]]],[.,.]]]
=> [[],[[[],[[],[[],[]]]],[[],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[[.,[[.,.],.]],[.,.]]]
=> [[],[[[],[[[],[]],[]]],[[],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> 0
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [[],[[],[[],[[],[[[],[]],[[],[]]]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [[],[[],[[],[[[],[]],[[],[[],[]]]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [[],[[],[[],[[[],[]],[[[],[]],[]]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [[],[[],[[],[[[],[[],[]]],[[],[]]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [[],[[],[[],[[[[],[]],[]],[[],[]]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [[],[[],[[],[[[[],[]],[[],[]]],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [[],[[],[[[],[]],[[],[[],[[],[]]]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [[],[[],[[[],[]],[[],[[[],[]],[]]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [[],[[],[[[],[]],[[[],[]],[[],[]]]]]]
=> ?
=> ?
=> ? = 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [[],[[],[[[],[]],[[[],[[],[]]],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [[],[[],[[[],[]],[[[[],[]],[]],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [[],[[],[[[],[[],[]]],[[],[[],[]]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [[],[[],[[[],[[],[]]],[[[],[]],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [[],[[],[[[[],[]],[]],[[],[[],[]]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [[],[[],[[[[],[]],[]],[[[],[]],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [[],[[],[[[],[[],[[],[]]]],[[],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [[],[[],[[[],[[[],[]],[]]],[[],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [[],[[],[[[[],[]],[[],[]]],[[],[]]]]]
=> ?
=> ?
=> ? = 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [[],[[],[[[[],[[],[]]],[]],[[],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [[],[[],[[[[[],[]],[]],[]],[[],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [[],[[],[[[],[[[],[]],[[],[]]]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [[],[[],[[[[],[]],[[],[[],[]]]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [[],[[],[[[[],[]],[[[],[]],[]]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [[],[[],[[[[],[[],[]]],[[],[]]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [[],[[],[[[[[],[]],[]],[[],[]]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [[],[[],[[[[[],[]],[[],[]]],[]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [[],[[[],[]],[[],[[],[[],[[],[]]]]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [[],[[[],[]],[[],[[],[[[],[]],[]]]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [[],[[[],[]],[[],[[[],[]],[[],[]]]]]]
=> ?
=> ?
=> ? = 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [[],[[[],[]],[[],[[[],[[],[]]],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [[],[[[],[]],[[],[[[[],[]],[]],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [[],[[[],[]],[[[],[]],[[],[[],[]]]]]]
=> ?
=> ?
=> ? = 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [[],[[[],[]],[[[],[]],[[[],[]],[]]]]]
=> ?
=> ?
=> ? = 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [[],[[[],[]],[[[],[[],[]]],[[],[]]]]]
=> ?
=> ?
=> ? = 1
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> [[],[[[],[]],[[[[],[]],[]],[[],[]]]]]
=> ?
=> ?
=> ? = 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [[],[[[],[]],[[[],[[],[[],[]]]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [[],[[[],[]],[[[],[[[],[]],[]]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [[],[[[],[]],[[[[],[]],[[],[]]],[]]]]
=> ?
=> ?
=> ? = 1
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [[],[[[],[]],[[[[],[[],[]]],[]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,.],[[[[.,.],.],.],.]]]
=> [[],[[[],[]],[[[[[],[]],[]],[]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [[],[[[],[[],[]]],[[],[[],[[],[]]]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [[],[[[],[[],[]]],[[],[[[],[]],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [[],[[[],[[],[]]],[[[],[]],[[],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [[],[[[],[[],[]]],[[[],[[],[]]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [[],[[[],[[],[]]],[[[[],[]],[]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> [[],[[[[],[]],[]],[[],[[],[[],[]]]]]]
=> ?
=> ?
=> ? = 0
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> [[],[[[[],[]],[]],[[],[[[],[]],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> [[],[[[[],[]],[]],[[[],[]],[[],[]]]]]
=> ?
=> ?
=> ? = 0
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [[],[[[[],[]],[]],[[[],[[],[]]],[]]]]
=> ?
=> ?
=> ? = 0
[.,[[[.,.],.],[[[.,.],.],.]]]
=> [[],[[[[],[]],[]],[[[[],[]],[]],[]]]]
=> ?
=> ?
=> ? = 0
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
Matching statistic: St001235
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> [2,1] => 2 = 0 + 2
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [3,1] => 2 = 0 + 2
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => 2 = 0 + 2
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => 2 = 0 + 2
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => 2 = 0 + 2
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> [3,1] => 2 = 0 + 2
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> [3,1] => 2 = 0 + 2
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [4,1] => 2 = 0 + 2
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 1 + 2
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 2 = 0 + 2
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 2 = 0 + 2
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 2 = 0 + 2
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2] => 2 = 0 + 2
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 3 = 1 + 2
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> [4,1] => 2 = 0 + 2
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> [4,1] => 2 = 0 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ([(3,5),(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ([(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1] => 3 = 1 + 2
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 2 = 0 + 2
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 2 = 0 + 2
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 2 = 0 + 2
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2] => 2 = 0 + 2
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> [5,1] => 2 = 0 + 2
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => ? = 1 + 2
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ([(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ([(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => ? = 1 + 2
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ([(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ([(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ([(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => ? = 1 + 2
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => ? = 1 + 2
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => ? = 1 + 2
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => ? = 1 + 2
[.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => ? = 1 + 2
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5,1,1] => ? = 1 + 2
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[[.,.],[[[[.,.],.],.],.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ([(2,6),(3,6),(4,6),(5,6)],7)
=> [6,1] => ? = 0 + 2
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ? = 0 + 2
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
[.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
[.,[[[.,.],.],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
[.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
=> [4,2,1] => ? = 0 + 2
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
[.,[[[.,.],.],[[[.,.],.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> [5,2] => ? = 0 + 2
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001568
Mp00013: Binary trees —to poset⟶ Posets
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 25%
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 25%
Values
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [3,2]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [7]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> ? = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> ? = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [8]
=> 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> 1 = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> 1 = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> 1 = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> ? = 1 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> 1 = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [5,3,3]
=> ? = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [5,3,3]
=> ? = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [5,3,3]
=> ? = 0 + 1
[.,[[[.,.],.],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> [5,3,3]
=> ? = 0 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> ? = 1 + 1
[.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> [6,4]
=> 1 = 0 + 1
[.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> 1 = 0 + 1
[.,[[[.,.],[.,[.,.]]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> 1 = 0 + 1
[.,[[[.,.],[[.,.],.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> 1 = 0 + 1
[.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> 1 = 0 + 1
[.,[[[[.,.],.],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> [9]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 1 + 1
[[.,.],[.,[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[.,.],[.,[[[.,.],.],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1 + 1
[[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1 + 1
[[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1 + 1
[[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1 + 1
[[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[.,.],[[.,[[.,.],.]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 1 + 1
[[.,.],[[[.,[.,.]],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 0 + 1
[[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[[.,.],.],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 0 + 1
[[[.,.],.],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[[.,.],.],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[.,[[.,.],.]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[.,[[.,.],.]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[[.,.],[.,.]],[.,[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 0 + 1
[[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> ? = 0 + 1
[[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[[[.,.],.],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[[[.,.],.],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> [13]
=> ? = 0 + 1
[[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[.,[.,[[.,.],.]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[.,[[.,.],[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> [5,4,2,2]
=> ? = 1 + 1
[[.,[[.,[.,.]],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[.,[[[.,.],.],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> [11]
=> ? = 0 + 1
[[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1 + 1
[[[.,.],[[.,.],.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1 + 1
[[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> [15]
=> ? = 1 + 1
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000257
Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 50%
Values
[[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7)
=> [3,2,1,1]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9)
=> [4,2,1,1,1]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> [4,2,1,1,1]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9)
=> [4,2,1,1,1]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> ([(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]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> ([(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]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> ([(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]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> ([(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]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> ([(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]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> ([(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]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> ([(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]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> ([(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]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> ([(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]
=> ? = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> ([(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]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> ([(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]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> ([(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]
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> ([(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]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> ([(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]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> ([(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]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> ([(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]
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> ([(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]
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> ([(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]
=> ? = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> ([(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]
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> ([(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]
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> ([(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]
=> 1 = 0 + 1
[[[.,.],[.,[.,.]]],.]
=> [[[[],[]],[[],[[],[]]]],[]]
=> ([(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]
=> 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
=> [[[[],[]],[[[],[]],[]]],[]]
=> ([(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]
=> 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
=> [[[[],[[],[]]],[[],[]]],[]]
=> ([(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]
=> 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
=> [[[[[],[]],[]],[[],[]]],[]]
=> ([(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]
=> 1 = 0 + 1
[[[[.,.],[.,.]],.],.]
=> [[[[[],[]],[[],[]]],[]],[]]
=> ([(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]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [[],[[],[[],[[[],[]],[[],[]]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [[],[[],[[[],[]],[[],[[],[]]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [[],[[],[[[],[]],[[[],[]],[]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [[],[[],[[[],[[],[]]],[[],[]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [[],[[],[[[[],[]],[]],[[],[]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [[],[[],[[[[],[]],[[],[]]],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [[],[[[],[]],[[],[[],[[],[]]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [[],[[[],[]],[[],[[[],[]],[]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [[],[[[],[]],[[[],[]],[[],[]]]]]
=> ([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> [5,2,2,1,1,1,1]
=> 2 = 1 + 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [[],[[[],[]],[[[],[[],[]]],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
=> [[],[[[],[]],[[[[],[]],[]],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [[],[[[],[[],[]]],[[],[[],[]]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> 1 = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [[],[[[],[[],[]]],[[[],[]],[]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [[],[[[[],[]],[]],[[],[[],[]]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> 1 = 0 + 1
[.,[[[.,.],.],[[.,.],.]]]
=> [[],[[[[],[]],[]],[[[],[]],[]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,11),(5,10),(6,9),(7,10),(8,11),(10,12),(11,12),(12,9)],13)
=> [5,3,1,1,1,1,1]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [[],[[[],[[],[[],[]]]],[[],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[.,[[.,.],.]],[.,.]]]
=> [[],[[[],[[[],[]],[]]],[[],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [[],[[[[],[]],[[],[]]],[[],[]]]]
=> ([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(11,12),(12,10)],13)
=> [5,2,2,1,1,1,1]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [[],[[[[],[[],[]]],[]],[[],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[[[.,.],.],.],[.,.]]]
=> [[],[[[[[],[]],[]],[]],[[],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[.,[[.,.],[.,.]]],.]]
=> [[],[[[],[[[],[]],[[],[]]]],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[[.,.],[.,[.,.]]],.]]
=> [[],[[[[],[]],[[],[[],[]]]],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[[.,.],[[.,.],.]],.]]
=> [[],[[[[],[]],[[[],[]],[]]],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[[.,[.,.]],[.,.]],.]]
=> [[],[[[[],[[],[]]],[[],[]]],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[[[.,.],.],[.,.]],.]]
=> [[],[[[[[],[]],[]],[[],[]]],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[.,[[[[.,.],[.,.]],.],.]]
=> [[],[[[[[],[]],[[],[]]],[]],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[[],[]],[[],[[],[[],[[],[]]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,.],[.,[.,[[.,.],.]]]]
=> [[[],[]],[[],[[],[[[],[]],[]]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,.],[.,[[.,.],[.,.]]]]
=> [[[],[]],[[],[[[],[]],[[],[]]]]]
=> ([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> [5,2,2,1,1,1,1]
=> 2 = 1 + 1
[[.,.],[.,[[.,[.,.]],.]]]
=> [[[],[]],[[],[[[],[[],[]]],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,.],[.,[[[.,.],.],.]]]
=> [[[],[]],[[],[[[[],[]],[]],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,.],[[.,.],[.,[.,.]]]]
=> [[[],[]],[[[],[]],[[],[[],[]]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> 2 = 1 + 1
[[.,.],[[.,.],[[.,.],.]]]
=> [[[],[]],[[[],[]],[[[],[]],[]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],[.,.]]]
=> [[[],[]],[[[],[[],[]]],[[],[]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],[.,.]]]
=> [[[],[]],[[[[],[]],[]],[[],[]]]]
=> ([(0,10),(1,7),(2,7),(3,8),(4,8),(5,9),(6,9),(7,12),(8,11),(9,10),(10,12),(12,11)],13)
=> [5,2,2,1,1,1,1]
=> 2 = 1 + 1
[[.,.],[[.,[.,[.,.]]],.]]
=> [[[],[]],[[[],[[],[[],[]]]],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,.],[[.,[[.,.],.]],.]]
=> [[[],[]],[[[],[[[],[]],[]]],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,.],[[[.,.],[.,.]],.]]
=> [[[],[]],[[[[],[]],[[],[]]],[]]]
=> ([(0,10),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> [5,2,2,1,1,1,1]
=> 2 = 1 + 1
[[.,.],[[[.,[.,.]],.],.]]
=> [[[],[]],[[[[],[[],[]]],[]],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,.],[[[[.,.],.],.],.]]
=> [[[],[]],[[[[[],[]],[]],[]],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,[.,.]],[.,[.,[.,.]]]]
=> [[[],[[],[]]],[[],[[],[[],[]]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> 1 = 0 + 1
[[.,[.,.]],[.,[[.,.],.]]]
=> [[[],[[],[]]],[[],[[[],[]],[]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> 1 = 0 + 1
[[.,[.,.]],[[.,.],[.,.]]]
=> [[[],[[],[]]],[[[],[]],[[],[]]]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
=> [4,3,2,1,1,1,1]
=> ? = 0 + 1
[[.,[.,.]],[[.,[.,.]],.]]
=> [[[],[[],[]]],[[[],[[],[]]],[]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> 1 = 0 + 1
[[.,[.,.]],[[[.,.],.],.]]
=> [[[],[[],[]]],[[[[],[]],[]],[]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,[.,.]]]]
=> [[[[],[]],[]],[[],[[],[[],[]]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> 1 = 0 + 1
[[[.,.],.],[.,[[.,.],.]]]
=> [[[[],[]],[]],[[],[[[],[]],[]]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],[.,.]]]
=> [[[[],[]],[]],[[[],[]],[[],[]]]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
=> [4,3,2,1,1,1,1]
=> ? = 0 + 1
[[[.,.],.],[[.,[.,.]],.]]
=> [[[[],[]],[]],[[[],[[],[]]],[]]]
=> ([(0,7),(1,7),(2,8),(3,8),(4,9),(5,10),(6,11),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
=> [5,3,1,1,1,1,1]
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,[.,.]]]
=> [[[[],[]],[[],[]]],[[],[[],[]]]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
=> [4,3,2,1,1,1,1]
=> ? = 0 + 1
[[[.,.],[.,.]],[[.,.],.]]
=> [[[[],[]],[[],[]]],[[[],[]],[]]]
=> ([(0,10),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,11),(9,10),(10,12),(11,12)],13)
=> [4,3,2,1,1,1,1]
=> ? = 0 + 1
[[.,[.,[.,[.,.]]]],[.,.]]
=> [[[],[[],[[],[[],[]]]]],[[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,[.,[[.,.],.]]],[.,.]]
=> [[[],[[],[[[],[]],[]]]],[[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,[[.,[.,.]],.]],[.,.]]
=> [[[],[[[],[[],[]]],[]]],[[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,[[[.,.],.],.]],[.,.]]
=> [[[],[[[[],[]],[]],[]]],[[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[[.,[.,[.,.]]],.],[.,.]]
=> [[[[],[[],[[],[]]]],[]],[[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[[.,[[.,.],.]],.],[.,.]]
=> [[[[],[[[],[]],[]]],[]],[[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[[[.,[.,.]],.],.],[.,.]]
=> [[[[[],[[],[]]],[]],[]],[[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[[[[.,.],.],.],.],[.,.]]
=> [[[[[[],[]],[]],[]],[]],[[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,11),(9,12),(10,9),(11,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,[.,[[.,.],[.,.]]]],.]
=> [[[],[[],[[[],[]],[[],[]]]]],[]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,[[.,.],[.,[.,.]]]],.]
=> [[[],[[[],[]],[[],[[],[]]]]],[]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,[[.,.],[[.,.],.]]],.]
=> [[[],[[[],[]],[[[],[]],[]]]],[]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,[[.,[.,.]],[.,.]]],.]
=> [[[],[[[],[[],[]]],[[],[]]]],[]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,[[[.,.],.],[.,.]]],.]
=> [[[],[[[[],[]],[]],[[],[]]]],[]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[.,[[[.,.],[.,.]],.]],.]
=> [[[],[[[[],[]],[[],[]]],[]]],[]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[[.,.],[.,[.,[.,.]]]],.]
=> [[[[],[]],[[],[[],[[],[]]]]],[]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
[[[.,.],[.,[[.,.],.]]],.]
=> [[[[],[]],[[],[[[],[]],[]]]],[]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,9),(5,10),(6,11),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
=> [6,2,1,1,1,1,1]
=> ? = 0 + 1
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
The following 60 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001618The cardinality of the Frattini sublattice of a lattice. St000741The Colin de Verdière graph invariant. St001597The Frobenius rank of a skew partition. St001545The second Elser number of a connected graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001638The book thickness of a graph. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001960The number of descents of a permutation minus one if its first entry is not one. St001729The number of visible descents of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001811The Castelnuovo-Mumford regularity of a permutation. St000023The number of inner peaks of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000353The number of inner valleys of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000779The tier of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000021The number of descents of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000897The number of different multiplicities of parts of an integer partition. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001812The biclique partition number of a graph. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000302The determinant of the distance matrix of a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000632The jump number of the poset. St000768The number of peaks in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001301The first Betti number of the order complex associated with the poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset.
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