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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000400
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(load all 2 compositions to match this statistic)
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000400: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00015: Binary trees —to ordered tree: right child = right brother⟶ Ordered trees
St000400: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[]]
=> [.,[.,.]]
=> [[],[]]
=> 2
[[[]]]
=> [[.,.],.]
=> [[[]]]
=> 3
[[],[],[]]
=> [.,[.,[.,.]]]
=> [[],[],[]]
=> 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [[],[[]]]
=> 4
[[[]],[]]
=> [[.,[.,.]],.]
=> [[[],[]]]
=> 5
[[[],[]]]
=> [[.,.],[.,.]]
=> [[[]],[]]
=> 4
[[[[]]]]
=> [[[.,.],.],.]
=> [[[[]]]]
=> 6
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [[],[],[],[]]
=> 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [[],[],[[]]]
=> 5
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [[],[[],[]]]
=> 6
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [[],[[]],[]]
=> 5
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [[],[[[]]]]
=> 7
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [[[],[],[]]]
=> 7
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [[[],[[]]]]
=> 8
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [[[],[]],[]]
=> 6
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [[[[],[]]]]
=> 9
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [[[]],[],[]]
=> 5
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [[[]],[[]]]
=> 6
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [[[[]]],[]]
=> 7
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [[[[]],[]]]
=> 8
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [[[[[]]]]]
=> 10
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [[],[],[],[],[]]
=> 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [[],[],[],[[]]]
=> 6
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [[],[],[[],[]]]
=> 7
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [[],[],[[]],[]]
=> 6
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [[],[],[[[]]]]
=> 8
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [[],[[],[],[]]]
=> 8
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [[],[[],[[]]]]
=> 9
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [[],[[],[]],[]]
=> 7
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [[],[[[],[]]]]
=> 10
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [[],[[]],[],[]]
=> 6
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [[],[[]],[[]]]
=> 7
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [[],[[[]]],[]]
=> 8
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [[],[[[]],[]]]
=> 9
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [[],[[[[]]]]]
=> 11
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [[[],[],[],[]]]
=> 9
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [[[],[],[[]]]]
=> 10
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [[[],[[],[]]]]
=> 11
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [[[]],[[]],[]]
=> 7
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [[[],[[[]]]]]
=> 12
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [[[],[]],[],[]]
=> 7
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [[[[],[],[]]]]
=> 12
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [[[],[]],[[]]]
=> 8
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [[[[],[[]]]]]
=> 13
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [[[],[],[]],[]]
=> 8
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [[[],[[]]],[]]
=> 9
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [[[[],[]]],[]]
=> 10
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [[[[],[]],[]]]
=> 11
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [[[[[],[]]]]]
=> 14
[[[],[],[],[]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [[[]],[],[],[]]
=> 6
Description
The path length of an ordered tree.
This is the sum of the lengths of all paths from the root to a node, see Section 2.3.4.5 of [1].
Matching statistic: St000639
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000639: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000639: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[],[]]
=> [.,[.,.]]
=> [2,1] => ([],2)
=> 2
[[[]]]
=> [[.,.],.]
=> [1,2] => ([(0,1)],2)
=> 3
[[],[],[]]
=> [.,[.,[.,.]]]
=> [3,2,1] => ([],3)
=> 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [2,3,1] => ([(1,2)],3)
=> 4
[[[]],[]]
=> [[.,[.,.]],.]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 5
[[[],[]]]
=> [[.,.],[.,.]]
=> [3,1,2] => ([(1,2)],3)
=> 4
[[[[]]]]
=> [[[.,.],.],.]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 6
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([],4)
=> 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(2,3)],4)
=> 5
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> 6
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [4,2,3,1] => ([(2,3)],4)
=> 5
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 7
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 7
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 8
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [4,2,1,3] => ([(1,3),(2,3)],4)
=> 6
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 9
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [4,3,1,2] => ([(2,3)],4)
=> 5
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 6
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 7
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 8
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 10
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => ([],5)
=> 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(3,4)],5)
=> 6
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(2,4),(3,4)],5)
=> 7
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => ([(3,4)],5)
=> 6
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(2,3),(3,4)],5)
=> 8
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
=> 8
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 9
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => ([(2,4),(3,4)],5)
=> 7
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> 10
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => ([(3,4)],5)
=> 6
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 7
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 8
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 9
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 11
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 9
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 10
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 11
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 7
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 12
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => ([(2,4),(3,4)],5)
=> 7
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
=> 12
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
=> 8
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 13
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
=> 8
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
=> 9
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
=> 10
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> 11
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 14
[[[],[],[],[]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 6
Description
The number of relations in a poset.
This is the number of intervals $x,y$ with $x\leq y$ in the poset, and therefore the dimension of the posets incidence algebra.
Matching statistic: St001228
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001228: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 80%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001228: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 80%
Values
[[],[]]
=> [.,[.,.]]
=> [1,0,1,0]
=> 2
[[[]]]
=> [[.,.],.]
=> [1,1,0,0]
=> 3
[[],[],[]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 4
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 5
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 4
[[[[]]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 6
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 5
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 6
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 5
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 7
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 7
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 8
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 6
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 9
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 5
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 6
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 7
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 8
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 10
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 6
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 7
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 6
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 8
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 8
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 9
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 7
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 10
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 6
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 7
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 8
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 9
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 11
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> 9
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> 10
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 11
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> 7
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> 12
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> 7
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> 12
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 8
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> 13
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> 8
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> 9
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> 10
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> 11
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> 14
[[[],[],[],[]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> 6
[[],[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[[],[],[],[],[],[[]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[[],[],[],[],[[]],[]]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9
[[],[],[],[],[[],[]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 8
[[],[],[],[],[[[]]]]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 10
[[],[],[],[[]],[],[]]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 10
[[],[],[],[[],[]],[]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 9
[[],[],[],[[[]]],[]]
=> [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 12
[[],[],[],[[],[],[]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 8
[[],[],[],[[],[[]]]]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 9
[[],[],[],[[[]],[]]]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 10
[[],[],[],[[[[]]]]]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 13
[[],[],[[]],[],[],[]]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 11
[[],[],[[]],[],[[]]]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 12
[[],[],[[]],[[]],[]]
=> [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 13
[[],[],[[]],[[],[]]]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 9
[[],[],[[],[]],[],[]]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 9
[[],[],[[[]]],[],[]]
=> [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 14
[[],[],[[],[]],[[]]]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 10
[[],[],[[],[],[]],[]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 10
[[],[],[[[]],[]],[]]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 12
[[],[],[[[],[]]],[]]
=> [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 13
[[],[],[[[[]]]],[]]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 16
[[],[],[[],[],[],[]]]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 8
[[],[],[[],[],[[]]]]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 9
[[],[],[[],[[]],[]]]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 10
[[],[],[[],[[],[]]]]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 12
[[],[],[[[]],[],[]]]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 10
[[],[],[[[[]]],[]]]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 13
[[],[],[[[],[],[]]]]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 12
[[],[[]],[],[],[],[]]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 12
[[],[[]],[],[],[[]]]
=> [.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 13
[[],[[]],[],[[]],[]]
=> [.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 14
[[],[[]],[],[[],[]]]
=> [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 9
[[],[[]],[],[[[]]]]
=> [.,[[.,[.,[[[.,.],.],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 15
[[],[[]],[[]],[],[]]
=> [.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 15
[[],[[]],[[],[]],[]]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 10
[[],[[]],[[[]]],[]]
=> [.,[[.,[[[.,[.,.]],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 17
[[],[[]],[[],[],[]]]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 9
[[],[[]],[[[[]]]]]
=> [.,[[.,[[[[.,.],.],.],.]],.]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 18
[[],[[],[]],[],[],[]]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 9
[[],[[[]]],[],[],[]]
=> [.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 16
[[],[[],[]],[],[[]]]
=> [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> ? = 10
[[],[[[]]],[],[[]]]
=> [.,[[[.,[.,[[.,.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 17
[[],[[],[]],[[]],[]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 11
[[],[[[]]],[[]],[]]
=> [.,[[[.,[[.,[.,.]],.]],.],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 18
[[],[[],[]],[[],[]]]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? = 10
[[],[[],[]],[[[]]]]
=> [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 12
[[],[[],[],[]],[],[]]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? = 10
[[],[[[]],[]],[],[]]
=> [.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? = 12
Description
The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000012
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 80%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 80%
Values
[[],[]]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[[]]]
=> [[.,.],.]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 3
[[],[],[]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[[[[]]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 7
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 6
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 9
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 6
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 8
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 6
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 6
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 8
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 8
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 9
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 7
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 10
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 6
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 7
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 9
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 11
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 9
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 10
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 11
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 7
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 12
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 7
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 12
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 8
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 13
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 8
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 9
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 10
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 11
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 14
[[[],[],[],[]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 6
[[],[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[[],[],[],[],[],[[]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 8
[[],[],[],[],[[]],[]]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 9
[[],[],[],[],[[],[]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 8
[[],[],[],[],[[[]]]]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 10
[[],[],[],[[]],[],[]]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 10
[[],[],[],[[],[]],[]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 9
[[],[],[],[[[]]],[]]
=> [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 12
[[],[],[],[[],[],[]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 8
[[],[],[],[[],[[]]]]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 9
[[],[],[],[[[]],[]]]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 10
[[],[],[],[[[[]]]]]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 13
[[],[],[[]],[],[],[]]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 11
[[],[],[[]],[],[[]]]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 12
[[],[],[[]],[[]],[]]
=> [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 13
[[],[],[[]],[[],[]]]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 9
[[],[],[[],[]],[],[]]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 9
[[],[],[[[]]],[],[]]
=> [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 14
[[],[],[[],[]],[[]]]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 10
[[],[],[[],[],[]],[]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 10
[[],[],[[[]],[]],[]]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 12
[[],[],[[[],[]]],[]]
=> [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 13
[[],[],[[[[]]]],[]]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 16
[[],[],[[],[],[],[]]]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 8
[[],[],[[],[],[[]]]]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 9
[[],[],[[],[[]],[]]]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 10
[[],[],[[],[[],[]]]]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 12
[[],[],[[[]],[],[]]]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 10
[[],[],[[[[]]],[]]]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 13
[[],[],[[[],[],[]]]]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 12
[[],[[]],[],[],[],[]]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 12
[[],[[]],[],[],[[]]]
=> [.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 13
[[],[[]],[],[[]],[]]
=> [.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 14
[[],[[]],[],[[],[]]]
=> [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 9
[[],[[]],[],[[[]]]]
=> [.,[[.,[.,[[[.,.],.],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 15
[[],[[]],[[]],[],[]]
=> [.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 15
[[],[[]],[[],[]],[]]
=> [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 10
[[],[[]],[[[]]],[]]
=> [.,[[.,[[[.,[.,.]],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> ? = 17
[[],[[]],[[],[],[]]]
=> [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 9
[[],[[]],[[[[]]]]]
=> [.,[[.,[[[[.,.],.],.],.]],.]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 18
[[],[[],[]],[],[],[]]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 9
[[],[[[]]],[],[],[]]
=> [.,[[[.,[.,[.,[.,.]]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 16
[[],[[],[]],[],[[]]]
=> [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 10
[[],[[[]]],[],[[]]]
=> [.,[[[.,[.,[[.,.],.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> ? = 17
[[],[[],[]],[[]],[]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 11
[[],[[[]]],[[]],[]]
=> [.,[[[.,[[.,[.,.]],.]],.],.]]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 18
[[],[[],[]],[[],[]]]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 10
[[],[[],[]],[[[]]]]
=> [.,[[.,[.,.]],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 12
[[],[[],[],[]],[],[]]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 10
[[],[[[]],[]],[],[]]
=> [.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 12
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St001295
Mp00139: Ordered trees —Zeilberger's Strahler bijection⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001295: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 56%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001295: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 56%
Values
[[],[]]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[[[]]]
=> [[.,.],.]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 3
[[],[],[]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[],[[]]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[[]],[]]
=> [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[[[],[]]]
=> [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[[[[]]]]
=> [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[[],[],[],[]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[],[],[[]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[[],[[]],[]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 6
[[],[[],[]]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
[[],[[[]]]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7
[[[]],[],[]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 7
[[[]],[[]]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 8
[[[],[]],[]]
=> [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 6
[[[[]]],[]]
=> [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 9
[[[],[],[]]]
=> [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[[[],[[]]]]
=> [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 6
[[[[]],[]]]
=> [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7
[[[[],[]]]]
=> [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 8
[[[[[]]]]]
=> [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10
[[],[],[],[],[]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[[],[],[],[[]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 6
[[],[],[[]],[]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 7
[[],[],[[],[]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 6
[[],[],[[[]]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 8
[[],[[]],[],[]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 8
[[],[[]],[[]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 9
[[],[[],[]],[]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 7
[[],[[[]]],[]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 10
[[],[[],[],[]]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 6
[[],[[],[[]]]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 7
[[],[[[]],[]]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
[[],[[[],[]]]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 9
[[],[[[[]]]]]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 11
[[[]],[],[],[]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 9
[[[]],[],[[]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 10
[[[]],[[]],[]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 11
[[[]],[[],[]]]
=> [[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 7
[[[]],[[[]]]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 12
[[[],[]],[],[]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 7
[[[[]]],[],[]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 12
[[[],[]],[[]]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 8
[[[[]]],[[]]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 13
[[[],[],[]],[]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 8
[[[],[[]]],[]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 9
[[[[]],[]],[]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 10
[[[[],[]]],[]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 11
[[[[[]]]],[]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 14
[[[],[],[],[]]]
=> [[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 6
[[],[],[],[],[],[]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[],[],[],[],[[]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 7
[[],[],[],[[]],[]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 8
[[],[],[],[[],[]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 7
[[],[],[],[[[]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 9
[[],[],[[]],[],[]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 9
[[],[],[[]],[[]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 10
[[],[],[[],[]],[]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 8
[[],[],[[[]]],[]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 11
[[],[],[[],[],[]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 7
[[],[],[[],[[]]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 8
[[],[],[[[]],[]]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 9
[[],[],[[[],[]]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 10
[[],[],[[[[]]]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 12
[[],[[]],[],[],[]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 10
[[],[[]],[],[[]]]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 11
[[],[[]],[[]],[]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 12
[[],[[]],[[],[]]]
=> [.,[[.,.],[[.,.],[.,.]]]]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 8
[[],[[]],[[[]]]]
=> [.,[[.,[[[.,.],.],.]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 13
[[],[[],[]],[],[]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 8
[[],[[[]]],[],[]]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 13
[[],[[],[]],[[]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 9
[[],[[[]]],[[]]]
=> [.,[[[.,[[.,.],.]],.],.]]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 14
[[],[[],[],[]],[]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 9
[[],[[],[[]]],[]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 10
[[],[[[]],[]],[]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 11
[[],[[[],[]]],[]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 12
[[],[[[[]]]],[]]
=> [.,[[[[.,[.,.]],.],.],.]]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 15
[[],[[],[],[],[]]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 7
[[],[[],[],[[]]]]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 8
[[],[[],[[]],[]]]
=> [.,[[.,.],[[.,[.,.]],.]]]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 9
[[],[[],[[],[]]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 11
[[],[[],[[[]]]]]
=> [.,[[.,.],[[[.,.],.],.]]]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 10
[[],[[[]],[],[]]]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 9
[[],[[[]],[[]]]]
=> [.,[[[.,.],.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 10
[[],[[[],[]],[]]]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 10
[[],[[[[]]],[]]]
=> [.,[[[[.,.],.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 12
[[],[[[],[],[]]]]
=> [.,[[[.,.],[.,[.,.]]],.]]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 11
[[],[[[],[[]]]]]
=> [.,[[[.,.],[[.,.],.]],.]]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 12
[[],[[[[]],[]]]]
=> [.,[[[[.,.],.],[.,.]],.]]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 13
[[],[[[[],[]]]]]
=> [.,[[[[.,.],[.,.]],.],.]]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 14
[[],[[[[[]]]]]]
=> [.,[[[[[.,.],.],.],.],.]]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 16
[[[]],[],[],[],[]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 11
[[[]],[],[],[[]]]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 12
[[[]],[],[[]],[]]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> ? = 13
[[[]],[],[[],[]]]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 8
[[[]],[],[[[]]]]
=> [[.,[.,[[[.,.],.],.]]],.]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 14
[[[]],[[]],[],[]]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 14
[[[]],[[]],[[]]]
=> [[.,[[.,[[.,.],.]],.]],.]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 15
[[[]],[[],[]],[]]
=> [[.,.],[[.,[.,.]],[.,.]]]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 9
Description
Gives the vector space dimension of the homomorphism space between J^2 and J^2.
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