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Your data matches 56 different statistics following compositions of up to 3 maps.
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Matching statistic: St001720
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(load all 4 compositions to match this statistic)
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> ([(0,1)],2)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> ([(0,2),(2,1)],3)
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(0,2),(2,1)],3)
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,3),(2,1),(3,2)],4)
=> 4
Description
The minimal length of a chain of small intervals in a lattice.
An interval $[a, b]$ is small if $b$ is a join of elements covering $a$.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 4
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[[.,.],.]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,.],[[.,[.,[.,.]]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,.],[[.,[[.,.],.]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,.],[[[[.,.],.],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[.,[[[.,.],.],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[[.,[.,[.,.]]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[[.,[[.,.],.]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ? = 6
[[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ? = 6
Description
The largest part of an integer partition.
Matching statistic: St000319
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> 1 = 2 - 1
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 2 = 3 - 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 2 = 3 - 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[.,[.,[[.,.],.]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[[.,[.,[.,.]]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[[.,[[.,.],.]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[[[[.,.],.],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,[[[.,.],.],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[[.,[.,[.,.]]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[[.,[[.,.],.]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ? = 6 - 1
[[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ? = 6 - 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> 0 = 1 - 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> 1 = 2 - 1
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 2 = 3 - 1
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> 2 = 3 - 1
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 4 = 5 - 1
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> 3 = 4 - 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[.,[.,[[.,.],.]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[[.,[.,[.,.]]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[[.,[[.,.],.]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,.],[[[[.,.],.],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[.,[[[.,.],.],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[[.,[.,[.,.]]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[[.,[[.,.],.]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ? = 7 - 1
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ? = 6 - 1
[[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ? = 6 - 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000010
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> [1]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> [1,1]
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [2,1]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,1,1]
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [2,1,1]
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,1,1,1,1]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 4
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[[.,.],.]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[.,[.,.]]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[[.,.],.]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[[.,.],.],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[[.,.],.],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[.,[.,.]]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[[.,.],.]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ?
=> ? = 6
[[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ?
=> ? = 6
Description
The length of the partition.
Matching statistic: St000676
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> [1,0]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[[.,.],.]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[.,[.,.]]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[[.,.],.]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[[.,.],.],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[[.,.],.],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[.,[.,.]]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[[.,.],.]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ?
=> ? = 6
[[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ?
=> ? = 6
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000734
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> [1]
=> [[1]]
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> [[1,2]]
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [[1,2,3]]
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [[1,2,3,4]]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[[.,.],.]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[.,[.,.]]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[[.,.],.]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[[.,.],.],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[[.,.],.],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[.,[.,.]]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[[.,.],.]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ?
=> ? = 6
[[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ?
=> ? = 6
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000097
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[[.,.],.]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[.,[.,.]]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[[.,.],.]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[[.,.],.],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[[.,.],.],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[.,[.,.]]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[[.,.],.]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ?
=> ? = 6
[[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ?
=> ? = 6
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000098
Values
[.,.]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[[.,.],.]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[.,[.,.]]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[[.,.],.]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[[.,.],.],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[[.,.],.],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[.,[.,.]]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[[.,.],.]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ?
=> ? = 6
[[.,.],[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ?
=> ? = 6
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St001039
Mp00013: Binary trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 88%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 88%
Values
[.,.]
=> ([],1)
=> [1]
=> [1,0]
=> ? = 1
[.,[.,.]]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[[.,.],.]
=> ([(0,1)],2)
=> [2]
=> [1,0,1,0]
=> 2
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> [3]
=> [1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,.],[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[.,[.,.]]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[.,[[.,.],.]]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[.,[.,.]],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[.,[[[.,.],.],.]],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[.,[.,.]]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[.,[[.,.],.]],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[.,[.,.]],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[.,[[[[[.,.],.],.],.],[.,.]]]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[.,[[.,.],.]]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[.,[.,.]],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[.,[[[.,.],.],.]]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[.,[.,.]]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[.,[[.,.],.]],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[.,[.,.]],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,.],[[[[.,.],.],.],.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[.,[[.,.],.]]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[.,[[[.,.],.],.]],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[.,[.,.]]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[.,[[.,.],.]],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[.,[.,.]],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[.,[[[[.,.],.],.],.]]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[.,[[[.,.],.],.]],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[.,[.,.]]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[.,[[.,.],.]],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[.,[.,.]],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[.,[[[[[.,.],.],.],.],.]]]
=> ([(0,7),(1,6),(2,7),(3,5),(4,3),(5,2),(6,4)],8)
=> ?
=> ?
=> ? = 7
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
=> ?
=> ?
=> ? = 6
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000744The length of the path to the largest entry in a standard Young tableau. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000808The number of up steps of the associated bargraph. St001494The Alon-Tarsi number of a graph. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000907The number of maximal antichains of minimal length in a poset. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001580The acyclic chromatic number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001963The tree-depth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St001644The dimension of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000050The depth or height of a binary tree. St001717The largest size of an interval in a poset. St001330The hat guessing number of a graph. St000080The rank of the poset. St000013The height of a Dyck path. St000822The Hadwiger number of the graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001323The independence gap of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001820The size of the image of the pop stack sorting operator. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001626The number of maximal proper sublattices of a lattice. St000299The number of nonisomorphic vertex-induced subtrees. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000225Difference between largest and smallest parts in a partition. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000442The maximal area to the right of an up step of a Dyck path. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one.
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