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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St000846
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Mp00047: Ordered trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[]
=> ([],1)
=> 0
[[]]
=> ([(0,1)],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 3
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 3
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 2
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 4
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 3
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> 2
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> 3
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> 2
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> 2
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000381
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 86% ●values known / values provided: 89%●distinct values known / distinct values provided: 86%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 86% ●values known / values provided: 89%●distinct values known / distinct values provided: 86%
Values
[]
=> .
=> ?
=> ? => ? = 0
[[]]
=> [.,.]
=> [1,0]
=> [1] => 1
[[],[]]
=> [[.,.],.]
=> [1,1,0,0]
=> [2] => 2
[[[]]]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1,1] => 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 3
[[],[[]]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 3
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 3
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 5
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => 4
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 4
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 3
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 3
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => 3
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 4
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 3
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 2
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 3
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 3
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 2
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 2
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 2
[[],[[],[[],[[],[[],[]]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[],[[],[[[],[]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[],[[[],[]],[[],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[],[[[],[[],[]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[],[[[[],[]],[]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[[],[]],[[],[[],[]]]]]
=> [[.,.],[[.,[[.,.],.]],[[.,.],[[.,.],.]]]]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[[],[]],[[[],[]],[]]]]
=> [[.,.],[[.,[[.,.],.]],[[.,[[.,.],.]],.]]]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[[],[[],[]]],[[],[]]]]
=> [[.,.],[[.,[[.,.],[[.,.],.]]],[[.,.],.]]]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[[[],[]],[]],[[],[]]]]
=> [[.,.],[[.,[[.,[[.,.],.]],.]],[[.,.],.]]]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[[],[[],[[],[]]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[[],[[[],[]],[]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[[[],[]],[[],[]]],[]]]
=> [[.,.],[[.,[[.,[[.,.],.]],[[.,.],.]]],.]]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[[[],[[],[]]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[],[[[[[],[]],[]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[]],[[],[[],[[],[]]]]]
=> [[.,[[.,.],.]],[[.,.],[[.,.],[[.,.],.]]]]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[]],[[],[[[],[]],[]]]]
=> [[.,[[.,.],.]],[[.,.],[[.,[[.,.],.]],.]]]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[]],[[[],[]],[[],[]]]]
=> [[.,[[.,.],.]],[[.,[[.,.],.]],[[.,.],.]]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[]],[[[],[[],[]]],[]]]
=> [[.,[[.,.],.]],[[.,[[.,.],[[.,.],.]]],.]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[]],[[[[],[]],[]],[]]]
=> [[.,[[.,.],.]],[[.,[[.,[[.,.],.]],.]],.]]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,.],[[.,.],.]]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,.],[[.,.],.]]],[[.,[[.,.],.]],.]]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[],[]],[]],[[],[[],[]]]]
=> [[.,[[.,[[.,.],.]],.]],[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[],[]],[]],[[[],[]],[]]]
=> [[.,[[.,[[.,.],.]],.]],[[.,[[.,.],.]],.]]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,.],[[.,.],[[.,.],.]]]],[[.,.],.]]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[.,.],[[.,[[.,.],.]],.]]],[[.,.],.]]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[],[]],[[],[]]],[[],[]]]
=> [[.,[[.,[[.,.],.]],[[.,.],.]]],[[.,.],.]]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[],[[],[]]],[]],[[],[]]]
=> [[.,[[.,[[.,.],[[.,.],.]]],.]],[[.,.],.]]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[[],[]],[]],[]],[[],[]]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],[[.,.],.]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,.],[[.,.],.]]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,[[.,.],.]],.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[.,.],[[.,[[.,.],.]],[[.,.],.]]]],.]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,.],[[.,.],.]]],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,[[.,.],.]],.]],.]]],.]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[],[]],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,.],.]],[[.,.],[[.,.],.]]]],.]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[],[]],[[[],[]],[]]],[]]
=> [[.,[[.,[[.,.],.]],[[.,[[.,.],.]],.]]],.]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[],[[],[]]],[[],[]]],[]]
=> [[.,[[.,[[.,.],[[.,.],.]]],[[.,.],.]]],.]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[[],[]],[]],[[],[]]],[]]
=> [[.,[[.,[[.,[[.,.],.]],.]],[[.,.],.]]],.]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[],[[],[[],[]]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,.],[[.,.],.]]]],.]],.]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[],[[[],[]],[]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,[[.,.],.]],.]]],.]],.]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[[],[]],[[],[]]],[]],[]]
=> [[.,[[.,[[.,[[.,.],.]],[[.,.],.]]],.]],.]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[[],[[],[]]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,.],[[.,.],.]]],.]],.]],.]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
[[[[[[],[]],[]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,[[.,.],.]],.]],.]],.]],.]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2] => ? = 2
Description
The largest part of an integer composition.
Matching statistic: St000392
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 86%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 86%
Values
[]
=> .
=> ? => ? => ? = 0 - 1
[[]]
=> [.,.]
=> [1] => => ? = 1 - 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => 1 => 1 = 2 - 1
[[[]]]
=> [[.,.],.]
=> [1,2] => 0 => 0 = 1 - 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 11 => 2 = 3 - 1
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 01 => 1 = 2 - 1
[[[]],[]]
=> [[.,.],[.,.]]
=> [1,3,2] => 01 => 1 = 2 - 1
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => 10 => 1 = 2 - 1
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => 00 => 0 = 1 - 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 111 => 3 = 4 - 1
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 011 => 2 = 3 - 1
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 011 => 2 = 3 - 1
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 101 => 1 = 2 - 1
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 001 => 1 = 2 - 1
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 011 => 2 = 3 - 1
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => 001 => 1 = 2 - 1
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 101 => 1 = 2 - 1
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => 001 => 1 = 2 - 1
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 110 => 2 = 3 - 1
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 010 => 1 = 2 - 1
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => 010 => 1 = 2 - 1
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 100 => 1 = 2 - 1
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 000 => 0 = 1 - 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1111 => 4 = 5 - 1
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 0111 => 3 = 4 - 1
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 0111 => 3 = 4 - 1
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1011 => 2 = 3 - 1
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 0011 => 2 = 3 - 1
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 0111 => 3 = 4 - 1
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 0011 => 2 = 3 - 1
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 1011 => 2 = 3 - 1
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 0011 => 2 = 3 - 1
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1101 => 2 = 3 - 1
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 0101 => 1 = 2 - 1
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 0101 => 1 = 2 - 1
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1001 => 1 = 2 - 1
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 0001 => 1 = 2 - 1
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 0111 => 3 = 4 - 1
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 0011 => 2 = 3 - 1
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 0011 => 2 = 3 - 1
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 0101 => 1 = 2 - 1
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 0001 => 1 = 2 - 1
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 1011 => 2 = 3 - 1
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 0011 => 2 = 3 - 1
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 1001 => 1 = 2 - 1
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 0001 => 1 = 2 - 1
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 1101 => 2 = 3 - 1
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 0101 => 1 = 2 - 1
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 0101 => 1 = 2 - 1
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 1001 => 1 = 2 - 1
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 0001 => 1 = 2 - 1
[[[],[],[],[]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 1110 => 3 = 4 - 1
[[],[[],[[],[[[],[]],[]]]]]
=> [.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]]
=> [5,4,7,6,8,3,9,2,10,1] => ? => ? = 2 - 1
[[],[[],[[[],[]],[[],[]]]]]
=> [.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]]
=> [4,3,7,6,8,5,9,2,10,1] => ? => ? = 2 - 1
[[],[[],[[[],[[],[]]],[]]]]
=> [.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]]
=> [5,4,6,3,8,7,9,2,10,1] => ? => ? = 2 - 1
[[],[[],[[[[],[]],[]],[]]]]
=> [.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,8,7,9,2,10,1] => ? => ? = 2 - 1
[[],[[[],[]],[[],[[],[]]]]]
=> [.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]]
=> [3,2,7,6,8,5,9,4,10,1] => ? => ? = 2 - 1
[[],[[[],[]],[[[],[]],[]]]]
=> [.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]]
=> [3,2,6,5,8,7,9,4,10,1] => ? => ? = 2 - 1
[[],[[[],[[],[]]],[[],[]]]]
=> [.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]]
=> [4,3,5,2,8,7,9,6,10,1] => ? => ? = 2 - 1
[[],[[[[],[]],[]],[[],[]]]]
=> [.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,5,4,8,7,9,6,10,1] => ? => ? = 2 - 1
[[],[[[],[[],[[],[]]]],[]]]
=> [.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]]
=> [5,4,6,3,7,2,9,8,10,1] => ? => ? = 2 - 1
[[],[[[],[[[],[]],[]]],[]]]
=> [.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]]
=> [4,3,6,5,7,2,9,8,10,1] => ? => ? = 2 - 1
[[],[[[[],[]],[[],[]]],[]]]
=> [.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]]
=> [3,2,6,5,7,4,9,8,10,1] => ? => ? = 2 - 1
[[],[[[[],[[],[]]],[]],[]]]
=> [.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]]
=> [4,3,5,2,7,6,9,8,10,1] => ? => ? = 2 - 1
[[[],[]],[[],[[],[[],[]]]]]
=> [[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [2,1,7,6,8,5,9,4,10,3] => ? => ? = 2 - 1
[[[],[]],[[],[[[],[]],[]]]]
=> [[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [2,1,6,5,8,7,9,4,10,3] => ? => ? = 2 - 1
[[[],[]],[[[],[]],[[],[]]]]
=> [[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [2,1,5,4,8,7,9,6,10,3] => ? => ? = 2 - 1
[[[],[]],[[[],[[],[]]],[]]]
=> [[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [2,1,6,5,7,4,9,8,10,3] => ? => ? = 2 - 1
[[[],[]],[[[[],[]],[]],[]]]
=> [[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [2,1,5,4,7,6,9,8,10,3] => ? => ? = 2 - 1
[[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]]
=> [3,2,4,1,8,7,9,6,10,5] => ? => ? = 2 - 1
[[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]]
=> [3,2,4,1,7,6,9,8,10,5] => ? => ? = 2 - 1
[[[[],[]],[]],[[],[[],[]]]]
=> [[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [2,1,4,3,8,7,9,6,10,5] => ? => ? = 2 - 1
[[[[],[]],[]],[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [2,1,4,3,7,6,9,8,10,5] => ? => ? = 2 - 1
[[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]]
=> [4,3,5,2,6,1,9,8,10,7] => ? => ? = 2 - 1
[[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,5,4,6,1,9,8,10,7] => ? => ? = 2 - 1
[[[[],[]],[[],[]]],[[],[]]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [2,1,5,4,6,3,9,8,10,7] => ? => ? = 2 - 1
[[[[],[[],[]]],[]],[[],[]]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]]
=> [3,2,4,1,6,5,9,8,10,7] => ? => ? = 2 - 1
[[[[[],[]],[]],[]],[[],[]]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]]
=> [2,1,4,3,6,5,9,8,10,7] => ? => ? = 2 - 1
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]]
=> [5,4,6,3,7,2,8,1,10,9] => ? => ? = 2 - 1
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]]
=> [4,3,6,5,7,2,8,1,10,9] => ? => ? = 2 - 1
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]
=> [3,2,6,5,7,4,8,1,10,9] => ? => ? = 2 - 1
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]]
=> [4,3,5,2,7,6,8,1,10,9] => ? => ? = 2 - 1
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,7,6,8,1,10,9] => ? => ? = 2 - 1
[[[[],[]],[[],[[],[]]]],[]]
=> [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [2,1,6,5,7,4,8,3,10,9] => ? => ? = 2 - 1
[[[[],[]],[[[],[]],[]]],[]]
=> [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [2,1,5,4,7,6,8,3,10,9] => ? => ? = 2 - 1
[[[[],[[],[]]],[[],[]]],[]]
=> [[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],[.,.]]
=> [3,2,4,1,7,6,8,5,10,9] => ? => ? = 2 - 1
[[[[[],[]],[]],[[],[]]],[]]
=> [[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],[.,.]]
=> [2,1,4,3,7,6,8,5,10,9] => ? => ? = 2 - 1
[[[[],[[],[[],[]]]],[]],[]]
=> [[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],[.,.]]
=> [4,3,5,2,6,1,8,7,10,9] => ? => ? = 2 - 1
[[[[],[[[],[]],[]]],[]],[]]
=> [[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],[.,.]]
=> [3,2,5,4,6,1,8,7,10,9] => ? => ? = 2 - 1
[[[[[],[]],[[],[]]],[]],[]]
=> [[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],[.,.]]
=> [2,1,5,4,6,3,8,7,10,9] => ? => ? = 2 - 1
[[],[[],[[],[[],[[],[[],[]]]]]]]
=> [.,[[.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]],.]]
=> [7,6,8,5,9,4,10,3,11,2,12,1] => ? => ? = 2 - 1
[[],[[],[[],[[],[[[],[]],[]]]]]]
=> [.,[[.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]],.]]
=> [6,5,8,7,9,4,10,3,11,2,12,1] => ? => ? = 2 - 1
[[],[[],[[],[[[],[]],[[],[]]]]]]
=> [.,[[.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]],.]]
=> [5,4,8,7,9,6,10,3,11,2,12,1] => ? => ? = 2 - 1
[[],[[],[[],[[[],[[],[]]],[]]]]]
=> [.,[[.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]],.]]
=> [6,5,7,4,9,8,10,3,11,2,12,1] => ? => ? = 2 - 1
[[],[[],[[],[[[[],[]],[]],[]]]]]
=> [.,[[.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]],.]]
=> [5,4,7,6,9,8,10,3,11,2,12,1] => ? => ? = 2 - 1
[[],[[],[[[],[]],[[],[[],[]]]]]]
=> [.,[[.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]],.]]
=> [4,3,8,7,9,6,10,5,11,2,12,1] => ? => ? = 2 - 1
[[],[[],[[[],[]],[[[],[]],[]]]]]
=> [.,[[.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]],.]]
=> [4,3,7,6,9,8,10,5,11,2,12,1] => ? => ? = 2 - 1
[[],[[],[[[],[[],[]]],[[],[]]]]]
=> [.,[[.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,9,8,10,7,11,2,12,1] => ? => ? = 2 - 1
[[],[[],[[[[],[]],[]],[[],[]]]]]
=> [.,[[.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]],.]]
=> [4,3,6,5,9,8,10,7,11,2,12,1] => ? => ? = 2 - 1
[[],[[],[[[],[[],[[],[]]]],[]]]]
=> [.,[[.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]],.]]
=> [6,5,7,4,8,3,10,9,11,2,12,1] => ? => ? = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000444
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 86%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 86%
Values
[]
=> []
=> []
=> ? = 0
[[]]
=> [1,0]
=> [1,0]
=> ? = 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[],[],[],[]]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[[],[[],[[],[[],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[],[[],[[[],[]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 2
[[],[[[],[]],[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 2
[[],[[[],[[],[]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
[[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2
[[[],[]],[[],[[],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 2
[[[],[[],[]]],[[],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 2
[[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 2
[[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 2
[[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> ? = 2
[[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 2
[[],[[[],[]],[[],[[],[]]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[[],[[[],[]],[[[],[]],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 2
[[],[[[],[[],[]]],[[],[]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
[[],[[[[],[]],[]],[[],[]]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2
[[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 2
[[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 2
[[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
[[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2
[[[],[]],[[],[[],[[],[]]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 2
[[[],[]],[[],[[[],[]],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[[[],[]],[[[],[]],[[],[]]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 2
[[[],[]],[[[],[[],[]]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[[[],[]],[[[[],[]],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 2
[[[],[[],[]]],[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 2
[[[],[[],[]]],[[[],[]],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 2
[[[[],[]],[]],[[],[[],[]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[[[[],[]],[]],[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[[[],[[],[[],[]]]],[[],[]]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
[[[],[[[],[]],[]]],[[],[]]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2
[[[[],[]],[[],[]]],[[],[]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> ? = 2
[[[[],[[],[]]],[]],[[],[]]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2
[[[[[],[]],[]],[]],[[],[]]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[[[],[[],[[],[[],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> ? = 2
[[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 2
[[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
[[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2
[[[[],[]],[[],[[],[]]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> ? = 2
[[[[],[]],[[[],[]],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 2
[[[[],[[],[]]],[[],[]]],[]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> ? = 2
[[[[[],[]],[]],[[],[]]],[]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> ? = 2
[[[[],[[],[[],[]]]],[]],[]]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 2
[[[[],[[[],[]],[]]],[]],[]]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St000845
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[]
=> ([],1)
=> ([],1)
=> 0
[[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[],[]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2)],3)
=> 2
[[[]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3)],4)
=> 3
[[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(3,1)],4)
=> 2
[[[],[]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[[[[]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 4
[[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
[[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
[[],[[],[]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[[],[[[]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 3
[[[]],[[]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> 2
[[[],[]],[]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
[[[[]]],[]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 2
[[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 3
[[[],[[]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[[[[]],[]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> 5
[[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 4
[[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 4
[[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 3
[[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> 3
[[],[[]],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 4
[[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> 3
[[],[[],[]],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 3
[[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> 3
[[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> 3
[[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> 2
[[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> 2
[[],[[[],[]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> 2
[[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 2
[[[]],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> 4
[[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> 3
[[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> 3
[[[]],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> 2
[[[]],[[[]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[[[],[]],[],[]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 3
[[[[]]],[],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ([(0,2),(0,3),(0,5),(4,1),(5,4)],6)
=> 3
[[[],[]],[[]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> 2
[[[[]]],[[]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> 2
[[[],[],[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> 3
[[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> 2
[[[[]],[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ([(0,3),(0,5),(4,2),(5,1),(5,4)],6)
=> 2
[[[[],[]]],[]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ([(0,3),(0,4),(4,5),(5,1),(5,2)],6)
=> 2
[[],[[],[[],[[],[]]]]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> ([(0,5),(0,8),(6,2),(6,7),(7,3),(7,4),(8,1),(8,6)],9)
=> ? = 2
[[],[[],[[[],[]],[]]]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> ([(0,5),(0,8),(6,2),(6,7),(7,3),(7,4),(8,1),(8,6)],9)
=> ? = 2
[[],[[[],[]],[[],[]]]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9)
=> ([(0,5),(0,8),(6,3),(6,4),(7,1),(7,2),(8,6),(8,7)],9)
=> ? = 2
[[],[[[],[[],[]]],[]]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> ([(0,5),(0,8),(6,2),(6,7),(7,3),(7,4),(8,1),(8,6)],9)
=> ? = 2
[[],[[[[],[]],[]],[]]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> ([(0,5),(0,8),(6,2),(6,7),(7,3),(7,4),(8,1),(8,6)],9)
=> ? = 2
[[[],[]],[[],[[],[]]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(0,8),(6,1),(6,2),(7,3),(7,4),(8,5),(8,6)],9)
=> ? = 2
[[[],[]],[[[],[]],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(0,8),(6,1),(6,2),(7,3),(7,4),(8,5),(8,6)],9)
=> ? = 2
[[[],[[],[]]],[[],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(0,8),(6,1),(6,2),(7,3),(7,4),(8,5),(8,6)],9)
=> ? = 2
[[[[],[]],[]],[[],[]]]
=> ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9)
=> ([(0,7),(0,8),(6,1),(6,2),(7,3),(7,4),(8,5),(8,6)],9)
=> ? = 2
[[[],[[],[[],[]]]],[]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> ([(0,5),(0,8),(6,2),(6,7),(7,3),(7,4),(8,1),(8,6)],9)
=> ? = 2
[[[],[[[],[]],[]]],[]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> ([(0,5),(0,8),(6,2),(6,7),(7,3),(7,4),(8,1),(8,6)],9)
=> ? = 2
[[[[],[]],[[],[]]],[]]
=> ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9)
=> ([(0,5),(0,8),(6,3),(6,4),(7,1),(7,2),(8,6),(8,7)],9)
=> ? = 2
[[[[],[[],[]]],[]],[]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> ([(0,5),(0,8),(6,2),(6,7),(7,3),(7,4),(8,1),(8,6)],9)
=> ? = 2
[[[[[],[]],[]],[]],[]]
=> ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9)
=> ([(0,5),(0,8),(6,2),(6,7),(7,3),(7,4),(8,1),(8,6)],9)
=> ? = 2
[[],[[],[[],[[],[[],[]]]]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[],[[],[[],[[[],[]],[]]]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[],[[],[[[],[]],[[],[]]]]]
=> ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,6),(0,10),(7,8),(7,9),(8,4),(8,5),(9,2),(9,3),(10,1),(10,7)],11)
=> ? = 2
[[],[[],[[[],[[],[]]],[]]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[],[[],[[[[],[]],[]],[]]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[],[[[],[]],[[],[[],[]]]]]
=> ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,6),(0,10),(7,1),(7,2),(8,3),(8,9),(9,4),(9,5),(10,7),(10,8)],11)
=> ? = 2
[[],[[[],[]],[[[],[]],[]]]]
=> ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,6),(0,10),(7,1),(7,2),(8,3),(8,9),(9,4),(9,5),(10,7),(10,8)],11)
=> ? = 2
[[],[[[],[[],[]]],[[],[]]]]
=> ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,6),(0,10),(7,1),(7,2),(8,3),(8,9),(9,4),(9,5),(10,7),(10,8)],11)
=> ? = 2
[[],[[[[],[]],[]],[[],[]]]]
=> ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,6),(0,10),(7,1),(7,2),(8,3),(8,9),(9,4),(9,5),(10,7),(10,8)],11)
=> ? = 2
[[],[[[],[[],[[],[]]]],[]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[],[[[],[[[],[]],[]]],[]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[],[[[[],[]],[[],[]]],[]]]
=> ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,6),(0,10),(7,8),(7,9),(8,4),(8,5),(9,2),(9,3),(10,1),(10,7)],11)
=> ? = 2
[[],[[[[],[[],[]]],[]],[]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[],[[[[[],[]],[]],[]],[]]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[[],[]],[[],[[],[[],[]]]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,9),(0,10),(7,4),(7,8),(8,5),(8,6),(9,2),(9,3),(10,1),(10,7)],11)
=> ? = 2
[[[],[]],[[],[[[],[]],[]]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,9),(0,10),(7,4),(7,8),(8,5),(8,6),(9,2),(9,3),(10,1),(10,7)],11)
=> ? = 2
[[[],[]],[[[],[]],[[],[]]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,9),(0,10),(7,5),(7,6),(8,3),(8,4),(9,1),(9,2),(10,7),(10,8)],11)
=> ? = 2
[[[],[]],[[[],[[],[]]],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,9),(0,10),(7,4),(7,8),(8,5),(8,6),(9,2),(9,3),(10,1),(10,7)],11)
=> ? = 2
[[[],[]],[[[[],[]],[]],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,9),(0,10),(7,4),(7,8),(8,5),(8,6),(9,2),(9,3),(10,1),(10,7)],11)
=> ? = 2
[[[],[[],[]]],[[],[[],[]]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> ([(0,9),(0,10),(7,3),(7,4),(8,5),(8,6),(9,2),(9,7),(10,1),(10,8)],11)
=> ? = 2
[[[],[[],[]]],[[[],[]],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> ([(0,9),(0,10),(7,3),(7,4),(8,5),(8,6),(9,2),(9,7),(10,1),(10,8)],11)
=> ? = 2
[[[[],[]],[]],[[],[[],[]]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> ([(0,9),(0,10),(7,3),(7,4),(8,5),(8,6),(9,2),(9,7),(10,1),(10,8)],11)
=> ? = 2
[[[[],[]],[]],[[[],[]],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11)
=> ([(0,9),(0,10),(7,3),(7,4),(8,5),(8,6),(9,2),(9,7),(10,1),(10,8)],11)
=> ? = 2
[[[],[[],[[],[]]]],[[],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,9),(0,10),(7,4),(7,8),(8,5),(8,6),(9,2),(9,3),(10,1),(10,7)],11)
=> ? = 2
[[[],[[[],[]],[]]],[[],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,9),(0,10),(7,4),(7,8),(8,5),(8,6),(9,2),(9,3),(10,1),(10,7)],11)
=> ? = 2
[[[[],[]],[[],[]]],[[],[]]]
=> ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,9),(0,10),(7,5),(7,6),(8,3),(8,4),(9,1),(9,2),(10,7),(10,8)],11)
=> ? = 2
[[[[],[[],[]]],[]],[[],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,9),(0,10),(7,4),(7,8),(8,5),(8,6),(9,2),(9,3),(10,1),(10,7)],11)
=> ? = 2
[[[[[],[]],[]],[]],[[],[]]]
=> ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,9),(0,10),(7,4),(7,8),(8,5),(8,6),(9,2),(9,3),(10,1),(10,7)],11)
=> ? = 2
[[[],[[],[[],[[],[]]]]],[]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[[],[[],[[[],[]],[]]]],[]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[[],[[[],[]],[[],[]]]],[]]
=> ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,6),(0,10),(7,8),(7,9),(8,4),(8,5),(9,2),(9,3),(10,1),(10,7)],11)
=> ? = 2
[[[],[[[],[[],[]]],[]]],[]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[[],[[[[],[]],[]],[]]],[]]
=> ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11)
=> ([(0,6),(0,10),(7,5),(7,9),(8,2),(8,3),(9,4),(9,8),(10,1),(10,7)],11)
=> ? = 2
[[[[],[]],[[],[[],[]]]],[]]
=> ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,6),(0,10),(7,1),(7,2),(8,3),(8,9),(9,4),(9,5),(10,7),(10,8)],11)
=> ? = 2
[[[[],[]],[[[],[]],[]]],[]]
=> ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,6),(0,10),(7,1),(7,2),(8,3),(8,9),(9,4),(9,5),(10,7),(10,8)],11)
=> ? = 2
[[[[],[[],[]]],[[],[]]],[]]
=> ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,6),(0,10),(7,1),(7,2),(8,3),(8,9),(9,4),(9,5),(10,7),(10,8)],11)
=> ? = 2
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000328
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
St000328: Ordered trees ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 86%
Values
[]
=> ? = 0
[[]]
=> 1
[[],[]]
=> 2
[[[]]]
=> 1
[[],[],[]]
=> 3
[[],[[]]]
=> 2
[[[]],[]]
=> 2
[[[],[]]]
=> 2
[[[[]]]]
=> 1
[[],[],[],[]]
=> 4
[[],[],[[]]]
=> 3
[[],[[]],[]]
=> 3
[[],[[],[]]]
=> 2
[[],[[[]]]]
=> 2
[[[]],[],[]]
=> 3
[[[]],[[]]]
=> 2
[[[],[]],[]]
=> 2
[[[[]]],[]]
=> 2
[[[],[],[]]]
=> 3
[[[],[[]]]]
=> 2
[[[[]],[]]]
=> 2
[[[[],[]]]]
=> 2
[[[[[]]]]]
=> 1
[[],[],[],[],[]]
=> 5
[[],[],[],[[]]]
=> 4
[[],[],[[]],[]]
=> 4
[[],[],[[],[]]]
=> 3
[[],[],[[[]]]]
=> 3
[[],[[]],[],[]]
=> 4
[[],[[]],[[]]]
=> 3
[[],[[],[]],[]]
=> 3
[[],[[[]]],[]]
=> 3
[[],[[],[],[]]]
=> 3
[[],[[],[[]]]]
=> 2
[[],[[[]],[]]]
=> 2
[[],[[[],[]]]]
=> 2
[[],[[[[]]]]]
=> 2
[[[]],[],[],[]]
=> 4
[[[]],[],[[]]]
=> 3
[[[]],[[]],[]]
=> 3
[[[]],[[],[]]]
=> 2
[[[]],[[[]]]]
=> 2
[[[],[]],[],[]]
=> 3
[[[[]]],[],[]]
=> 3
[[[],[]],[[]]]
=> 2
[[[[]]],[[]]]
=> 2
[[[],[],[]],[]]
=> 3
[[[],[[]]],[]]
=> 2
[[[[]],[]],[]]
=> 2
[[[[],[]]],[]]
=> 2
[[[[[]]]],[]]
=> 2
[[],[[],[[],[[],[]]]]]
=> ? = 2
[[],[[],[[[],[]],[]]]]
=> ? = 2
[[],[[[],[]],[[],[]]]]
=> ? = 2
[[],[[[],[[],[]]],[]]]
=> ? = 2
[[],[[[[],[]],[]],[]]]
=> ? = 2
[[[],[]],[[],[[],[]]]]
=> ? = 2
[[[],[]],[[[],[]],[]]]
=> ? = 2
[[[],[[],[]]],[[],[]]]
=> ? = 2
[[[[],[]],[]],[[],[]]]
=> ? = 2
[[[],[[],[[],[]]]],[]]
=> ? = 2
[[[],[[[],[]],[]]],[]]
=> ? = 2
[[[[],[]],[[],[]]],[]]
=> ? = 2
[[[[],[[],[]]],[]],[]]
=> ? = 2
[[[[[],[]],[]],[]],[]]
=> ? = 2
[[],[[],[[],[[],[[],[]]]]]]
=> ? = 2
[[],[[],[[],[[[],[]],[]]]]]
=> ? = 2
[[],[[],[[[],[]],[[],[]]]]]
=> ? = 2
[[],[[],[[[],[[],[]]],[]]]]
=> ? = 2
[[],[[],[[[[],[]],[]],[]]]]
=> ? = 2
[[],[[[],[]],[[],[[],[]]]]]
=> ? = 2
[[],[[[],[]],[[[],[]],[]]]]
=> ? = 2
[[],[[[],[[],[]]],[[],[]]]]
=> ? = 2
[[],[[[[],[]],[]],[[],[]]]]
=> ? = 2
[[],[[[],[[],[[],[]]]],[]]]
=> ? = 2
[[],[[[],[[[],[]],[]]],[]]]
=> ? = 2
[[],[[[[],[]],[[],[]]],[]]]
=> ? = 2
[[],[[[[],[[],[]]],[]],[]]]
=> ? = 2
[[],[[[[[],[]],[]],[]],[]]]
=> ? = 2
[[[],[]],[[],[[],[[],[]]]]]
=> ? = 2
[[[],[]],[[],[[[],[]],[]]]]
=> ? = 2
[[[],[]],[[[],[]],[[],[]]]]
=> ? = 2
[[[],[]],[[[],[[],[]]],[]]]
=> ? = 2
[[[],[]],[[[[],[]],[]],[]]]
=> ? = 2
[[[],[[],[]]],[[],[[],[]]]]
=> ? = 2
[[[],[[],[]]],[[[],[]],[]]]
=> ? = 2
[[[[],[]],[]],[[],[[],[]]]]
=> ? = 2
[[[[],[]],[]],[[[],[]],[]]]
=> ? = 2
[[[],[[],[[],[]]]],[[],[]]]
=> ? = 2
[[[],[[[],[]],[]]],[[],[]]]
=> ? = 2
[[[[],[]],[[],[]]],[[],[]]]
=> ? = 2
[[[[],[[],[]]],[]],[[],[]]]
=> ? = 2
[[[[[],[]],[]],[]],[[],[]]]
=> ? = 2
[[[],[[],[[],[[],[]]]]],[]]
=> ? = 2
[[[],[[],[[[],[]],[]]]],[]]
=> ? = 2
[[[],[[[],[]],[[],[]]]],[]]
=> ? = 2
[[[],[[[],[[],[]]],[]]],[]]
=> ? = 2
[[[],[[[[],[]],[]],[]]],[]]
=> ? = 2
[[[[],[]],[[],[[],[]]]],[]]
=> ? = 2
[[[[],[]],[[[],[]],[]]],[]]
=> ? = 2
Description
The maximum number of child nodes in a tree.
Matching statistic: St000308
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00049: Ordered trees —to binary tree: left brother = left child⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 86%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 86%
Values
[]
=> .
=> ? => ? = 0
[[]]
=> [.,.]
=> [1] => 1
[[],[]]
=> [[.,.],.]
=> [1,2] => 2
[[[]]]
=> [.,[.,.]]
=> [2,1] => 1
[[],[],[]]
=> [[[.,.],.],.]
=> [1,2,3] => 3
[[],[[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => 2
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => 2
[[[],[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => 2
[[[[]]]]
=> [.,[.,[.,.]]]
=> [3,2,1] => 1
[[],[],[],[]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => 4
[[],[],[[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[[],[[]],[]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => 3
[[],[[],[]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[],[[[]]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[[[]],[],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => 3
[[[]],[[]]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => 2
[[[],[]],[]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[[[[]]],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2
[[[],[],[]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[[[],[[]]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2
[[[[]],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[[[[],[]]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[[[[[]]]]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[[],[],[],[],[]]
=> [[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => 5
[[],[],[],[[]]]
=> [[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 4
[[],[],[[]],[]]
=> [[[[.,.],.],[.,.]],.]
=> [4,1,2,3,5] => 4
[[],[],[[],[]]]
=> [[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 3
[[],[],[[[]]]]
=> [[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 3
[[],[[]],[],[]]
=> [[[[.,.],[.,.]],.],.]
=> [3,1,2,4,5] => 4
[[],[[]],[[]]]
=> [[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 3
[[],[[],[]],[]]
=> [[[.,.],[[.,.],.]],.]
=> [3,4,1,2,5] => 3
[[],[[[]]],[]]
=> [[[.,.],[.,[.,.]]],.]
=> [4,3,1,2,5] => 3
[[],[[],[],[]]]
=> [[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 3
[[],[[],[[]]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[[],[[[]],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[[],[[[],[]]]]
=> [[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[[],[[[[]]]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 2
[[[]],[],[],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => 4
[[[]],[],[[]]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 3
[[[]],[[]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => 3
[[[]],[[],[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[]],[[[]]]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2
[[[],[]],[],[]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => 3
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => 3
[[[],[]],[[]]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 2
[[[[]]],[[]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 2
[[[],[],[]],[]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => 3
[[[],[[]]],[]]
=> [[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => 2
[[[[]],[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => 2
[[[[],[]]],[]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => 2
[[[[[]]]],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => 2
[[],[[],[[],[[],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> [7,8,5,6,3,4,1,2] => ? = 2
[[],[[],[[[],[]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],.]]]
=> [6,7,5,8,3,4,1,2] => ? = 2
[[],[[[],[]],[[],[]]]]
=> [[.,.],[[.,[[.,.],.]],[[.,.],.]]]
=> [7,8,4,5,3,6,1,2] => ? = 2
[[],[[[],[[],[]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,.],.]]],.]]
=> [6,7,4,5,3,8,1,2] => ? = 2
[[],[[[[],[]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,.],.]],.]],.]]
=> [5,6,4,7,3,8,1,2] => ? = 2
[[[],[]],[[],[[],[]]]]
=> [[.,[[.,.],.]],[[.,.],[[.,.],.]]]
=> [7,8,5,6,2,3,1,4] => ? = 2
[[[],[]],[[[],[]],[]]]
=> [[.,[[.,.],.]],[[.,[[.,.],.]],.]]
=> [6,7,5,8,2,3,1,4] => ? = 2
[[[],[[],[]]],[[],[]]]
=> [[.,[[.,.],[[.,.],.]]],[[.,.],.]]
=> [7,8,4,5,2,3,1,6] => ? = 2
[[[[],[]],[]],[[],[]]]
=> [[.,[[.,[[.,.],.]],.]],[[.,.],.]]
=> [7,8,3,4,2,5,1,6] => ? = 2
[[[],[[],[[],[]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,.],.]]]],.]
=> [6,7,4,5,2,3,1,8] => ? = 2
[[[],[[[],[]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,.],.]],.]]],.]
=> [5,6,4,7,2,3,1,8] => ? = 2
[[[[],[]],[[],[]]],[]]
=> [[.,[[.,[[.,.],.]],[[.,.],.]]],.]
=> [6,7,3,4,2,5,1,8] => ? = 2
[[[[],[[],[]]],[]],[]]
=> [[.,[[.,[[.,.],[[.,.],.]]],.]],.]
=> [5,6,3,4,2,7,1,8] => ? = 2
[[[[[],[]],[]],[]],[]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],.]
=> [4,5,3,6,2,7,1,8] => ? = 2
[[],[[],[[],[[],[[],[]]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,.],[[.,.],.]]]]]
=> [9,10,7,8,5,6,3,4,1,2] => ? = 2
[[],[[],[[],[[[],[]],[]]]]]
=> [[.,.],[[.,.],[[.,.],[[.,[[.,.],.]],.]]]]
=> [8,9,7,10,5,6,3,4,1,2] => ? = 2
[[],[[],[[[],[]],[[],[]]]]]
=> [[.,.],[[.,.],[[.,[[.,.],.]],[[.,.],.]]]]
=> [9,10,6,7,5,8,3,4,1,2] => ? = 2
[[],[[],[[[],[[],[]]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,.],[[.,.],.]]],.]]]
=> [8,9,6,7,5,10,3,4,1,2] => ? = 2
[[],[[],[[[[],[]],[]],[]]]]
=> [[.,.],[[.,.],[[.,[[.,[[.,.],.]],.]],.]]]
=> [7,8,6,9,5,10,3,4,1,2] => ? = 2
[[],[[[],[]],[[],[[],[]]]]]
=> [[.,.],[[.,[[.,.],.]],[[.,.],[[.,.],.]]]]
=> [9,10,7,8,4,5,3,6,1,2] => ? = 2
[[],[[[],[]],[[[],[]],[]]]]
=> [[.,.],[[.,[[.,.],.]],[[.,[[.,.],.]],.]]]
=> [8,9,7,10,4,5,3,6,1,2] => ? = 2
[[],[[[],[[],[]]],[[],[]]]]
=> [[.,.],[[.,[[.,.],[[.,.],.]]],[[.,.],.]]]
=> [9,10,6,7,4,5,3,8,1,2] => ? = 2
[[],[[[[],[]],[]],[[],[]]]]
=> [[.,.],[[.,[[.,[[.,.],.]],.]],[[.,.],.]]]
=> [9,10,5,6,4,7,3,8,1,2] => ? = 2
[[],[[[],[[],[[],[]]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,.],[[.,.],.]]]],.]]
=> [8,9,6,7,4,5,3,10,1,2] => ? = 2
[[],[[[],[[[],[]],[]]],[]]]
=> [[.,.],[[.,[[.,.],[[.,[[.,.],.]],.]]],.]]
=> [7,8,6,9,4,5,3,10,1,2] => ? = 2
[[],[[[[],[]],[[],[]]],[]]]
=> [[.,.],[[.,[[.,[[.,.],.]],[[.,.],.]]],.]]
=> [8,9,5,6,4,7,3,10,1,2] => ? = 2
[[],[[[[],[[],[]]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,.],[[.,.],.]]],.]],.]]
=> [7,8,5,6,4,9,3,10,1,2] => ? = 2
[[],[[[[[],[]],[]],[]],[]]]
=> [[.,.],[[.,[[.,[[.,[[.,.],.]],.]],.]],.]]
=> [6,7,5,8,4,9,3,10,1,2] => ? = 2
[[[],[]],[[],[[],[[],[]]]]]
=> [[.,[[.,.],.]],[[.,.],[[.,.],[[.,.],.]]]]
=> [9,10,7,8,5,6,2,3,1,4] => ? = 2
[[[],[]],[[],[[[],[]],[]]]]
=> [[.,[[.,.],.]],[[.,.],[[.,[[.,.],.]],.]]]
=> [8,9,7,10,5,6,2,3,1,4] => ? = 2
[[[],[]],[[[],[]],[[],[]]]]
=> [[.,[[.,.],.]],[[.,[[.,.],.]],[[.,.],.]]]
=> [9,10,6,7,5,8,2,3,1,4] => ? = 2
[[[],[]],[[[],[[],[]]],[]]]
=> [[.,[[.,.],.]],[[.,[[.,.],[[.,.],.]]],.]]
=> [8,9,6,7,5,10,2,3,1,4] => ? = 2
[[[],[]],[[[[],[]],[]],[]]]
=> [[.,[[.,.],.]],[[.,[[.,[[.,.],.]],.]],.]]
=> [7,8,6,9,5,10,2,3,1,4] => ? = 2
[[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,.],[[.,.],.]]],[[.,.],[[.,.],.]]]
=> [9,10,7,8,4,5,2,3,1,6] => ? = 2
[[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,.],[[.,.],.]]],[[.,[[.,.],.]],.]]
=> [8,9,7,10,4,5,2,3,1,6] => ? = 2
[[[[],[]],[]],[[],[[],[]]]]
=> [[.,[[.,[[.,.],.]],.]],[[.,.],[[.,.],.]]]
=> [9,10,7,8,3,4,2,5,1,6] => ? = 2
[[[[],[]],[]],[[[],[]],[]]]
=> [[.,[[.,[[.,.],.]],.]],[[.,[[.,.],.]],.]]
=> [8,9,7,10,3,4,2,5,1,6] => ? = 2
[[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,.],[[.,.],[[.,.],.]]]],[[.,.],.]]
=> [9,10,6,7,4,5,2,3,1,8] => ? = 2
[[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[.,.],[[.,[[.,.],.]],.]]],[[.,.],.]]
=> [9,10,5,6,4,7,2,3,1,8] => ? = 2
[[[[],[]],[[],[]]],[[],[]]]
=> [[.,[[.,[[.,.],.]],[[.,.],.]]],[[.,.],.]]
=> [9,10,6,7,3,4,2,5,1,8] => ? = 2
[[[[],[[],[]]],[]],[[],[]]]
=> [[.,[[.,[[.,.],[[.,.],.]]],.]],[[.,.],.]]
=> [9,10,5,6,3,4,2,7,1,8] => ? = 2
[[[[[],[]],[]],[]],[[],[]]]
=> [[.,[[.,[[.,[[.,.],.]],.]],.]],[[.,.],.]]
=> [9,10,4,5,3,6,2,7,1,8] => ? = 2
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,.],[[.,.],.]]]]],.]
=> [8,9,6,7,4,5,2,3,1,10] => ? = 2
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,.],[[.,.],[[.,[[.,.],.]],.]]]],.]
=> [7,8,6,9,4,5,2,3,1,10] => ? = 2
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[.,.],[[.,[[.,.],.]],[[.,.],.]]]],.]
=> [8,9,5,6,4,7,2,3,1,10] => ? = 2
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,.],[[.,.],.]]],.]]],.]
=> [7,8,5,6,4,9,2,3,1,10] => ? = 2
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[.,.],[[.,[[.,[[.,.],.]],.]],.]]],.]
=> [6,7,5,8,4,9,2,3,1,10] => ? = 2
[[[[],[]],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,.],.]],[[.,.],[[.,.],.]]]],.]
=> [8,9,6,7,3,4,2,5,1,10] => ? = 2
[[[[],[]],[[[],[]],[]]],[]]
=> [[.,[[.,[[.,.],.]],[[.,[[.,.],.]],.]]],.]
=> [7,8,6,9,3,4,2,5,1,10] => ? = 2
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices {0,1,2,…,n} and root 0 in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St001235
Mp00050: Ordered trees —to binary tree: right brother = right child⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 86%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 86%
Values
[]
=> .
=> ? => ? => ? = 0
[[]]
=> [.,.]
=> [1] => [1] => 1
[[],[]]
=> [.,[.,.]]
=> [2,1] => [1,1] => 2
[[[]]]
=> [[.,.],.]
=> [1,2] => [2] => 1
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => [1,1,1] => 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => [2,1] => 2
[[[]],[]]
=> [[.,.],[.,.]]
=> [1,3,2] => [2,1] => 2
[[[],[]]]
=> [[.,[.,.]],.]
=> [2,1,3] => [1,2] => 2
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => [3] => 1
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1] => 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => [2,1,1] => 3
[[],[[]],[]]
=> [.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,1,1] => 3
[[],[[],[]]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,1] => 2
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,1] => 2
[[[]],[],[]]
=> [[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,1] => 3
[[[]],[[]]]
=> [[.,.],[[.,.],.]]
=> [1,3,4,2] => [3,1] => 2
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,2,1] => 2
[[[[]]],[]]
=> [[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,1] => 2
[[[],[],[]]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,2] => 3
[[[],[[]]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,2] => 2
[[[[]],[]]]
=> [[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,2] => 2
[[[[],[]]]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3] => 2
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => [4] => 1
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1] => 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [2,1,1,1] => 4
[[],[],[[]],[]]
=> [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [2,1,1,1] => 4
[[],[],[[],[]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,1,1] => 3
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,1,1] => 3
[[],[[]],[],[]]
=> [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,1,1,1] => 4
[[],[[]],[[]]]
=> [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,1,1] => 3
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,2,1,1] => 3
[[],[[[]]],[]]
=> [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,1,1] => 3
[[],[[],[],[]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,2,1] => 3
[[],[[],[[]]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,2,1] => 2
[[],[[[]],[]]]
=> [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,2,1] => 2
[[],[[[],[]]]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,3,1] => 2
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,1] => 2
[[[]],[],[],[]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,1,1,1] => 4
[[[]],[],[[]]]
=> [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,1,1] => 3
[[[]],[[]],[]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,1,1] => 3
[[[]],[[],[]]]
=> [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,2,1] => 2
[[[]],[[[]]]]
=> [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [4,1] => 2
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [1,2,1,1] => 3
[[[[]]],[],[]]
=> [[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,1,1] => 3
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,3,1] => 2
[[[[]]],[[]]]
=> [[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [4,1] => 2
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [1,1,2,1] => 3
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,2,1] => 2
[[[[]],[]],[]]
=> [[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,2,1] => 2
[[[[],[]]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,1] => 2
[[[[[]]]],[]]
=> [[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,1] => 2
[[],[[],[[],[[],[]]]]]
=> [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [1,2,2,2,1] => ? = 2
[[],[[],[[[],[]],[]]]]
=> [.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,7,2,8,1] => [1,2,2,2,1] => ? = 2
[[],[[[],[]],[[],[]]]]
=> [.,[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,6,5,7,4,8,1] => [1,2,2,2,1] => ? = 2
[[],[[[],[[],[]]],[]]]
=> [.,[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [4,3,5,2,7,6,8,1] => [1,2,2,2,1] => ? = 2
[[],[[[[],[]],[]],[]]]
=> [.,[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,8,1] => [1,2,2,2,1] => ? = 2
[[[],[]],[[],[[],[]]]]
=> [[.,[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [2,1,6,5,7,4,8,3] => [1,2,2,2,1] => ? = 2
[[[],[]],[[[],[]],[]]]
=> [[.,[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [2,1,5,4,7,6,8,3] => [1,2,2,2,1] => ? = 2
[[[],[[],[]]],[[],[]]]
=> [[.,[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,4,1,7,6,8,5] => [1,2,2,2,1] => ? = 2
[[[[],[]],[]],[[],[]]]
=> [[[.,[.,.]],[.,.]],[[.,[.,.]],.]]
=> [2,1,4,3,7,6,8,5] => [1,2,2,2,1] => ? = 2
[[[],[[],[[],[]]]],[]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [4,3,5,2,6,1,8,7] => [1,2,2,2,1] => ? = 2
[[[],[[[],[]],[]]],[]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,6,1,8,7] => [1,2,2,2,1] => ? = 2
[[[[],[]],[[],[]]],[]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[.,.]]
=> [2,1,5,4,6,3,8,7] => [1,2,2,2,1] => ? = 2
[[[[],[[],[]]],[]],[]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[.,.]]
=> [3,2,4,1,6,5,8,7] => [1,2,2,2,1] => ? = 2
[[[[[],[]],[]],[]],[]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [2,1,4,3,6,5,8,7] => [1,2,2,2,1] => ? = 2
[[],[[],[[],[[],[[],[]]]]]]
=> [.,[[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],.]]
=> [6,5,7,4,8,3,9,2,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[],[[],[[[],[]],[]]]]]
=> [.,[[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],.]]
=> [5,4,7,6,8,3,9,2,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[],[[[],[]],[[],[]]]]]
=> [.,[[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],.]]
=> [4,3,7,6,8,5,9,2,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[],[[[],[[],[]]],[]]]]
=> [.,[[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],.]]
=> [5,4,6,3,8,7,9,2,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[],[[[[],[]],[]],[]]]]
=> [.,[[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],.]]
=> [4,3,6,5,8,7,9,2,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[[],[]],[[],[[],[]]]]]
=> [.,[[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],.]]
=> [3,2,7,6,8,5,9,4,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[[],[]],[[[],[]],[]]]]
=> [.,[[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],.]]
=> [3,2,6,5,8,7,9,4,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[[],[[],[]]],[[],[]]]]
=> [.,[[[.,[[.,[.,.]],.]],[[.,[.,.]],.]],.]]
=> [4,3,5,2,8,7,9,6,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[[[],[]],[]],[[],[]]]]
=> [.,[[[[.,[.,.]],[.,.]],[[.,[.,.]],.]],.]]
=> [3,2,5,4,8,7,9,6,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[[],[[],[[],[]]]],[]]]
=> [.,[[[.,[[.,[[.,[.,.]],.]],.]],[.,.]],.]]
=> [5,4,6,3,7,2,9,8,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[[],[[[],[]],[]]],[]]]
=> [.,[[[.,[[[.,[.,.]],[.,.]],.]],[.,.]],.]]
=> [4,3,6,5,7,2,9,8,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[[[],[]],[[],[]]],[]]]
=> [.,[[[[.,[.,.]],[[.,[.,.]],.]],[.,.]],.]]
=> [3,2,6,5,7,4,9,8,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[[[],[[],[]]],[]],[]]]
=> [.,[[[[.,[[.,[.,.]],.]],[.,.]],[.,.]],.]]
=> [4,3,5,2,7,6,9,8,10,1] => [1,2,2,2,2,1] => ? = 2
[[],[[[[[],[]],[]],[]],[]]]
=> [.,[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],.]]
=> [3,2,5,4,7,6,9,8,10,1] => [1,2,2,2,2,1] => ? = 2
[[[],[]],[[],[[],[[],[]]]]]
=> [[.,[.,.]],[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [2,1,7,6,8,5,9,4,10,3] => [1,2,2,2,2,1] => ? = 2
[[[],[]],[[],[[[],[]],[]]]]
=> [[.,[.,.]],[[.,[[[.,[.,.]],[.,.]],.]],.]]
=> [2,1,6,5,8,7,9,4,10,3] => [1,2,2,2,2,1] => ? = 2
[[[],[]],[[[],[]],[[],[]]]]
=> [[.,[.,.]],[[[.,[.,.]],[[.,[.,.]],.]],.]]
=> [2,1,5,4,8,7,9,6,10,3] => [1,2,2,2,2,1] => ? = 2
[[[],[]],[[[],[[],[]]],[]]]
=> [[.,[.,.]],[[[.,[[.,[.,.]],.]],[.,.]],.]]
=> [2,1,6,5,7,4,9,8,10,3] => [1,2,2,2,2,1] => ? = 2
[[[],[]],[[[[],[]],[]],[]]]
=> [[.,[.,.]],[[[[.,[.,.]],[.,.]],[.,.]],.]]
=> [2,1,5,4,7,6,9,8,10,3] => [1,2,2,2,2,1] => ? = 2
[[[],[[],[]]],[[],[[],[]]]]
=> [[.,[[.,[.,.]],.]],[[.,[[.,[.,.]],.]],.]]
=> [3,2,4,1,8,7,9,6,10,5] => [1,2,2,2,2,1] => ? = 2
[[[],[[],[]]],[[[],[]],[]]]
=> [[.,[[.,[.,.]],.]],[[[.,[.,.]],[.,.]],.]]
=> [3,2,4,1,7,6,9,8,10,5] => [1,2,2,2,2,1] => ? = 2
[[[[],[]],[]],[[],[[],[]]]]
=> [[[.,[.,.]],[.,.]],[[.,[[.,[.,.]],.]],.]]
=> [2,1,4,3,8,7,9,6,10,5] => [1,2,2,2,2,1] => ? = 2
[[[[],[]],[]],[[[],[]],[]]]
=> [[[.,[.,.]],[.,.]],[[[.,[.,.]],[.,.]],.]]
=> [2,1,4,3,7,6,9,8,10,5] => [1,2,2,2,2,1] => ? = 2
[[[],[[],[[],[]]]],[[],[]]]
=> [[.,[[.,[[.,[.,.]],.]],.]],[[.,[.,.]],.]]
=> [4,3,5,2,6,1,9,8,10,7] => [1,2,2,2,2,1] => ? = 2
[[[],[[[],[]],[]]],[[],[]]]
=> [[.,[[[.,[.,.]],[.,.]],.]],[[.,[.,.]],.]]
=> [3,2,5,4,6,1,9,8,10,7] => [1,2,2,2,2,1] => ? = 2
[[[[],[]],[[],[]]],[[],[]]]
=> [[[.,[.,.]],[[.,[.,.]],.]],[[.,[.,.]],.]]
=> [2,1,5,4,6,3,9,8,10,7] => [1,2,2,2,2,1] => ? = 2
[[[[],[[],[]]],[]],[[],[]]]
=> [[[.,[[.,[.,.]],.]],[.,.]],[[.,[.,.]],.]]
=> [3,2,4,1,6,5,9,8,10,7] => [1,2,2,2,2,1] => ? = 2
[[[[[],[]],[]],[]],[[],[]]]
=> [[[[.,[.,.]],[.,.]],[.,.]],[[.,[.,.]],.]]
=> [2,1,4,3,6,5,9,8,10,7] => [1,2,2,2,2,1] => ? = 2
[[[],[[],[[],[[],[]]]]],[]]
=> [[.,[[.,[[.,[[.,[.,.]],.]],.]],.]],[.,.]]
=> [5,4,6,3,7,2,8,1,10,9] => [1,2,2,2,2,1] => ? = 2
[[[],[[],[[[],[]],[]]]],[]]
=> [[.,[[.,[[[.,[.,.]],[.,.]],.]],.]],[.,.]]
=> [4,3,6,5,7,2,8,1,10,9] => [1,2,2,2,2,1] => ? = 2
[[[],[[[],[]],[[],[]]]],[]]
=> [[.,[[[.,[.,.]],[[.,[.,.]],.]],.]],[.,.]]
=> [3,2,6,5,7,4,8,1,10,9] => [1,2,2,2,2,1] => ? = 2
[[[],[[[],[[],[]]],[]]],[]]
=> [[.,[[[.,[[.,[.,.]],.]],[.,.]],.]],[.,.]]
=> [4,3,5,2,7,6,8,1,10,9] => [1,2,2,2,2,1] => ? = 2
[[[],[[[[],[]],[]],[]]],[]]
=> [[.,[[[[.,[.,.]],[.,.]],[.,.]],.]],[.,.]]
=> [3,2,5,4,7,6,8,1,10,9] => [1,2,2,2,2,1] => ? = 2
[[[[],[]],[[],[[],[]]]],[]]
=> [[[.,[.,.]],[[.,[[.,[.,.]],.]],.]],[.,.]]
=> [2,1,6,5,7,4,8,3,10,9] => [1,2,2,2,2,1] => ? = 2
[[[[],[]],[[[],[]],[]]],[]]
=> [[[.,[.,.]],[[[.,[.,.]],[.,.]],.]],[.,.]]
=> [2,1,5,4,7,6,8,3,10,9] => [1,2,2,2,2,1] => ? = 2
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001239
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 86%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 86%
Values
[]
=> []
=> []
=> []
=> ? = 0
[[]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[[],[[],[[],[[],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[],[[[],[]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[],[]],[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[],[[],[]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[[],[]],[[],[[],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[],[]]],[[],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[],[[],[]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[[],[]],[]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[],[]],[[],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[],[[],[]]],[]],[]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[[],[]],[]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[],[]],[[],[[],[]]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[],[]],[[[],[]],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[],[[],[]]],[[],[]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[[],[]],[]],[[],[]]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 2
[[[],[]],[[],[[],[[],[]]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[]],[[],[[[],[]],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[]],[[[],[]],[[],[]]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[]],[[[],[[],[]]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[]],[[[[],[]],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[],[]]],[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[],[]]],[[[],[]],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[],[]],[]],[[],[[],[]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[],[]],[]],[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[],[[],[]]]],[[],[]]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[[],[]],[]]],[[],[]]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[],[]],[[],[]]],[[],[]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[],[[],[]]],[]],[[],[]]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[[],[]],[]],[]],[[],[]]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[],[[],[[],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[],[]],[[],[[],[]]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
[[[[],[]],[[[],[]],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001372
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 86%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 86%
Values
[]
=> []
=> []
=> => ? = 0
[[]]
=> [1,0]
=> [1,0]
=> 10 => 1
[[],[]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1100 => 2
[[[]]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1010 => 1
[[],[],[]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 3
[[],[[]]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 2
[[[]],[]]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 110010 => 2
[[[],[]]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 2
[[[[]]]]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 1
[[],[],[],[]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4
[[],[],[[]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3
[[],[[]],[]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3
[[],[[],[]]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 2
[[],[[[]]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 2
[[[]],[],[]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => 3
[[[]],[[]]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 2
[[[],[]],[]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 2
[[[[]]],[]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 2
[[[],[],[]]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3
[[[],[[]]]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 2
[[[[]],[]]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 2
[[[[],[]]]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 2
[[[[[]]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 1
[[],[],[],[],[]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1111100000 => 5
[[],[],[],[[]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 4
[[],[],[[]],[]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 4
[[],[],[[],[]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1110110000 => 3
[[],[],[[[]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 3
[[],[[]],[],[]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 4
[[],[[]],[[]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 3
[[],[[],[]],[]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1110011000 => 3
[[],[[[]]],[]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 3
[[],[[],[],[]]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 3
[[],[[],[[]]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 2
[[],[[[]],[]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 2
[[],[[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 2
[[],[[[[]]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 2
[[[]],[],[],[]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 4
[[[]],[],[[]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 3
[[[]],[[]],[]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 3
[[[]],[[],[]]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 2
[[[]],[[[]]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 2
[[[],[]],[],[]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 3
[[[[]]],[],[]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 3
[[[],[]],[[]]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 2
[[[[]]],[[]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 2
[[[],[],[]],[]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 3
[[[],[[]]],[]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 2
[[[[]],[]],[]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 2
[[[[],[]]],[]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 2
[[[[[]]]],[]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 2
[[],[[],[[],[[],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> 1101101101100000 => ? = 2
[[],[[],[[[],[]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> 1101101100110000 => ? = 2
[[],[[[],[]],[[],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> 1101101100011000 => ? = 2
[[],[[[],[[],[]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> 1101100110110000 => ? = 2
[[],[[[[],[]],[]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 1101100110011000 => ? = 2
[[[],[]],[[],[[],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> 1101101100001100 => ? = 2
[[[],[]],[[[],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> 1101100110001100 => ? = 2
[[[],[[],[]]],[[],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> 1101100011011000 => ? = 2
[[[[],[]],[]],[[],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 1101100011001100 => ? = 2
[[[],[[],[[],[]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> 1100110110110000 => ? = 2
[[[],[[[],[]],[]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> 1100110110011000 => ? = 2
[[[[],[]],[[],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> 1100110110001100 => ? = 2
[[[[],[[],[]]],[]],[]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> 1100110011011000 => ? = 2
[[[[[],[]],[]],[]],[]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 1100110011001100 => ? = 2
[[],[[],[[],[[],[[],[]]]]]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> 11011011011011000000 => ? = 2
[[],[[],[[],[[[],[]],[]]]]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
=> 11011011011001100000 => ? = 2
[[],[[],[[[],[]],[[],[]]]]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,0]
=> 11011011011000110000 => ? = 2
[[],[[],[[[],[[],[]]],[]]]]
=> [1,0,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
=> 11011011001101100000 => ? = 2
[[],[[],[[[[],[]],[]],[]]]]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> 11011011001100110000 => ? = 2
[[],[[[],[]],[[],[[],[]]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> 11011011011000011000 => ? = 2
[[],[[[],[]],[[[],[]],[]]]]
=> [1,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> 11011011001100011000 => ? = 2
[[],[[[],[[],[]]],[[],[]]]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0,0]
=> 11011011000110110000 => ? = 2
[[],[[[[],[]],[]],[[],[]]]]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> 11011011000110011000 => ? = 2
[[],[[[],[[],[[],[]]]],[]]]
=> [1,0,1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> 11011001101101100000 => ? = 2
[[],[[[],[[[],[]],[]]],[]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> 11011001101100110000 => ? = 2
[[],[[[[],[]],[[],[]]],[]]]
=> [1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> 11011001101100011000 => ? = 2
[[],[[[[],[[],[]]],[]],[]]]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> 11011001100110110000 => ? = 2
[[],[[[[[],[]],[]],[]],[]]]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 11011001100110011000 => ? = 2
[[[],[]],[[],[[],[[],[]]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> 11011011011000001100 => ? = 2
[[[],[]],[[],[[[],[]],[]]]]
=> [1,1,0,1,0,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> 11011011001100001100 => ? = 2
[[[],[]],[[[],[]],[[],[]]]]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
=> 11011011000110001100 => ? = 2
[[[],[]],[[[],[[],[]]],[]]]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> 11011001101100001100 => ? = 2
[[[],[]],[[[[],[]],[]],[]]]
=> [1,1,0,1,0,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> 11011001100110001100 => ? = 2
[[[],[[],[]]],[[],[[],[]]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,1,1,0,0,0]
=> 11011011000011011000 => ? = 2
[[[],[[],[]]],[[[],[]],[]]]
=> [1,1,0,1,1,0,1,0,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> 11011001100011011000 => ? = 2
[[[[],[]],[]],[[],[[],[]]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> 11011011000011001100 => ? = 2
[[[[],[]],[]],[[[],[]],[]]]
=> [1,1,1,0,1,0,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 11011001100011001100 => ? = 2
[[[],[[],[[],[]]]],[[],[]]]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> 11011000110110110000 => ? = 2
[[[],[[[],[]],[]]],[[],[]]]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> 11011000110110011000 => ? = 2
[[[[],[]],[[],[]]],[[],[]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> 11011000110110001100 => ? = 2
[[[[],[[],[]]],[]],[[],[]]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> 11011000110011011000 => ? = 2
[[[[[],[]],[]],[]],[[],[]]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 11011000110011001100 => ? = 2
[[[],[[],[[],[[],[]]]]],[]]
=> [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> 11001101101101100000 => ? = 2
[[[],[[],[[[],[]],[]]]],[]]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> 11001101101100110000 => ? = 2
[[[],[[[],[]],[[],[]]]],[]]
=> [1,1,0,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> 11001101101100011000 => ? = 2
[[[],[[[],[[],[]]],[]]],[]]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> 11001101100110110000 => ? = 2
[[[],[[[[],[]],[]],[]]],[]]
=> [1,1,0,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 11001101100110011000 => ? = 2
[[[[],[]],[[],[[],[]]]],[]]
=> [1,1,1,0,1,0,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> 11001101101100001100 => ? = 2
[[[[],[]],[[[],[]],[]]],[]]
=> [1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> 11001101100110001100 => ? = 2
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001192The maximal dimension of Ext2A(S,A) for a simple module S over the corresponding Nakayama algebra A. St000485The length of the longest cycle of a permutation. St001062The maximal size of a block of a set partition. St001330The hat guessing number of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
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