Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000776
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00223: Permutations runsortPermutations
Mp00160: Permutations graph of inversionsGraphs
St000776: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => ([],1)
=> 1
[.,[.,.]]
=> [2,1] => [1,2] => ([],2)
=> 2
[[.,.],.]
=> [1,2] => [1,2] => ([],2)
=> 2
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => ([],3)
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => ([],3)
=> 3
[[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => ([],3)
=> 3
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => ([(1,2)],3)
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => ([],3)
=> 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => ([],4)
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => ([],4)
=> 4
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => ([],4)
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => ([],4)
=> 4
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => ([],4)
=> 4
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => ([],4)
=> 4
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => ([],4)
=> 4
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => ([(2,3)],4)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => ([],5)
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => ([],5)
=> 5
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => ([],5)
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => ([],5)
=> 5
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => ([],5)
=> 5
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => ([],5)
=> 5
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => ([(3,4)],5)
=> 3
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => ([],5)
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => ([],5)
=> 5
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => ([],5)
=> 5
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => ([],5)
=> 5
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => ([],5)
=> 5
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => ([(3,4)],5)
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => ([],5)
=> 5
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => ([(3,4)],5)
=> 3
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => ([(3,4)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => ([],5)
=> 5
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => ([],5)
=> 5
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 3
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => ([(3,4)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 3
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => ([],5)
=> 5
Description
The maximal multiplicity of an eigenvalue in a graph.
Mp00018: Binary trees left border symmetryBinary trees
Mp00013: Binary trees to posetPosets
Mp00198: Posets incomparability graphGraphs
St000986: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ([],1)
=> 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ([],2)
=> 2
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ([],2)
=> 2
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> 3
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> 2
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 3
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 3
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 3
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 3
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 3
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 3
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> 3
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> 3
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(2,4),(3,4)],5)
=> 3
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> 3
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 5
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St001641
Mp00012: Binary trees to Dyck path: up step, left tree, down step, right treeDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
Mp00215: Set partitions Wachs-WhiteSet partitions
St001641: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> {{1}}
=> {{1}}
=> 0 = 1 - 1
[.,[.,.]]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 1 = 2 - 1
[[.,.],.]
=> [1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 2 = 3 - 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 2 = 3 - 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 2 = 3 - 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 0 = 1 - 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 3 = 4 - 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 3 = 4 - 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 3 = 4 - 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 3 = 4 - 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 3 = 4 - 1
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 3 = 4 - 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,3},{2,4}}
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> 4 = 5 - 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> 4 = 5 - 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> 2 = 3 - 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> 4 = 5 - 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> 4 = 5 - 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 4 = 5 - 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> 2 = 3 - 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> 2 = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3},{2,4},{5}}
=> 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 4 = 5 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 4 = 5 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 4 = 5 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 4 = 5 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 4 = 5 - 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 4 = 5 - 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 4 = 5 - 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> 2 = 3 - 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> {{1},{2,4},{3,5}}
=> 2 = 3 - 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 4 = 5 - 1
Description
The number of ascent tops in the flattened set partition such that all smaller elements appear before. Let $P$ be a set partition. The flattened set partition is the permutation obtained by sorting the set of blocks of $P$ according to their minimal element and the elements in each block in increasing order. Given a set partition $P$, this statistic is the binary logarithm of the number of set partitions that flatten to the same permutation as $P$.
Mp00018: Binary trees left border symmetryBinary trees
St000385: Binary trees ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> ? = 1 - 1
[.,[.,.]]
=> [.,[.,.]]
=> 1 = 2 - 1
[[.,.],.]
=> [[.,.],.]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> 2 = 3 - 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> 2 = 3 - 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 2 = 3 - 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> 0 = 1 - 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> 3 = 4 - 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> 3 = 4 - 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> 3 = 4 - 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> 3 = 4 - 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 3 = 4 - 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> 4 = 5 - 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> 4 = 5 - 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> 2 = 3 - 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> 4 = 5 - 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> 4 = 5 - 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> 4 = 5 - 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> 2 = 3 - 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> 2 = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> 4 = 5 - 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> 4 = 5 - 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> 4 = 5 - 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> 4 = 5 - 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 4 = 5 - 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> 4 = 5 - 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> 2 = 3 - 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> 4 = 5 - 1
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> 2 = 3 - 1
Description
The number of vertices with out-degree 1 in a binary tree. See the references for several connections of this statistic. In particular, the number $T(n,k)$ of binary trees with $n$ vertices and $k$ out-degree $1$ vertices is given by $T(n,k) = 0$ for $n-k$ odd and $$T(n,k)=\frac{2^k}{n+1}\binom{n+1}{k}\binom{n+1-k}{(n-k)/2}$$ for $n-k$ is even.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00223: Permutations runsortPermutations
Mp00066: Permutations inversePermutations
St000696: Permutations ⟶ ℤResult quality: 97% values known / values provided: 97%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 2 = 1 + 1
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 3 = 2 + 1
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 3 = 2 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 4 = 3 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [1,2,3] => 4 = 3 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2,3] => [1,2,3] => 4 = 3 + 1
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 4 = 3 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 5 = 4 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 5 = 4 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 5 = 4 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 5 = 4 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 5 = 4 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 5 = 4 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 5 = 4 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => [1,3,4,2] => 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => [1,3,4,2] => 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [1,4,2,3] => 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 5 = 4 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => [1,2,4,5,3] => 4 = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,3,4] => 4 = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,2,3,5,4] => [1,2,3,5,4] => 4 = 3 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,3,2,4,5] => [1,3,2,4,5] => 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 4 = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,4,2,3,5] => [1,3,4,2,5] => 4 = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,2,4,3,5] => [1,2,4,3,5] => 4 = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,3,4,2,5] => [1,4,2,3,5] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4,5] => [1,2,3,4,5] => 6 = 5 + 1
[[[[.,[.,.]],.],.],[.,[.,.]]]
=> [7,6,2,1,3,4,5] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 5 + 1
[[[[.,[.,.]],.],.],[[.,.],.]]
=> [6,7,2,1,3,4,5] => [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ? = 5 + 1
[[[[.,[.,.]],.],[.,.]],[.,.]]
=> [7,5,2,1,3,4,6] => [1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => ? = 3 + 1
[[[[.,[.,.]],[.,.]],.],[.,.]]
=> [7,4,2,1,3,5,6] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => ? = 3 + 1
[[[[.,[.,[.,.]]],.],.],[.,.]]
=> [7,3,2,1,4,5,6] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 5 + 1
[[[[.,[[.,.],.]],.],.],[.,.]]
=> [7,2,3,1,4,5,6] => [1,4,5,6,2,3,7] => [1,5,6,2,3,4,7] => ? = 5 + 1
[[[[[.,[.,.]],.],.],.],[.,.]]
=> [7,2,1,3,4,5,6] => [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ? = 5 + 1
[[[[.,[.,.]],.],[.,[.,.]]],.]
=> [6,5,2,1,3,4,7] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 3 + 1
[[[[.,[.,.]],.],[[.,.],.]],.]
=> [5,6,2,1,3,4,7] => [1,3,4,7,2,5,6] => [1,5,2,3,6,7,4] => ? = 3 + 1
[[[[.,[.,.]],[.,.]],[.,.]],.]
=> [6,4,2,1,3,5,7] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 1 + 1
[[[[.,[.,[.,.]]],.],[.,.]],.]
=> [6,3,2,1,4,5,7] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 3 + 1
[[[[.,[[.,.],.]],.],[.,.]],.]
=> [6,2,3,1,4,5,7] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 3 + 1
[[[.,[[.,[.,.]],[.,.]]],.],.]
=> [5,3,2,4,1,6,7] => [1,6,7,2,4,3,5] => [1,4,6,5,7,2,3] => ? = 3 + 1
[[[.,[[.,[.,[.,.]]],.]],.],.]
=> [4,3,2,5,1,6,7] => [1,6,7,2,5,3,4] => [1,4,6,7,5,2,3] => ? = 3 + 1
[[[.,[[.,[[.,.],.]],.]],.],.]
=> [3,4,2,5,1,6,7] => [1,6,7,2,5,3,4] => [1,4,6,7,5,2,3] => ? = 3 + 1
[[[.,[[[.,[.,.]],.],.]],.],.]
=> [3,2,4,5,1,6,7] => [1,6,7,2,4,5,3] => [1,4,7,5,6,2,3] => ? = 3 + 1
[[[[.,[.,.]],[.,[.,.]]],.],.]
=> [5,4,2,1,3,6,7] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 3 + 1
[[[[.,[.,.]],[[.,.],.]],.],.]
=> [4,5,2,1,3,6,7] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => ? = 3 + 1
[[[[.,[[.,[.,.]],.]],.],.],.]
=> [3,2,4,1,5,6,7] => [1,5,6,7,2,4,3] => [1,5,7,6,2,3,4] => ? = 3 + 1
Description
The number of cycles in the breakpoint graph of a permutation. The breakpoint graph of a permutation $\pi_1,\dots,\pi_n$ is the directed, bicoloured graph with vertices $0,\dots,n$, a grey edge from $i$ to $i+1$ and a black edge from $\pi_i$ to $\pi_{i-1}$ for $0\leq i\leq n$, all indices taken modulo $n+1$. This graph decomposes into alternating cycles, which this statistic counts. The distribution of this statistic on permutations of $n-1$ is, according to [cor.1, 5] and [eq.6, 6], given by $$ \frac{1}{n(n+1)}((q+n)_{n+1}-(q)_{n+1}), $$ where $(x)_n=x(x-1)\dots(x-n+1)$.
Mp00018: Binary trees left border symmetryBinary trees
Mp00013: Binary trees to posetPosets
St001631: Posets ⟶ ℤResult quality: 42% values known / values provided: 42%distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> ([],1)
=> 0 = 1 - 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> 1 = 2 - 1
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 3 - 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 3 - 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 3 - 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 5 - 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 5 - 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ? = 3 - 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [.,[.,[[[.,.],[[.,.],.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ? = 3 - 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [.,[.,[[[[.,.],[.,.]],.],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [.,[.,[[[[.,.],.],[.,.]],.]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [.,[[.,[.,[[.,.],[.,.]]]],.]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 5 - 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [.,[[.,[[.,[.,.]],[.,.]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
=> [.,[[.,[[.,.],[.,[.,.]]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[[.,.],[[.,[[.,.],.]],.]]]
=> [.,[[.,[[.,.],[[.,.],.]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[[.,.],[[[.,.],[.,.]],.]]]
=> [.,[[.,[[[.,.],[.,.]],.]],.]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 5 - 1
[.,[[.,.],[[[.,[.,.]],.],.]]]
=> [.,[[.,[[[.,.],.],[.,.]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 5 - 1
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 5 - 1
[.,[[.,[.,.]],[.,[[.,.],.]]]]
=> [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 5 - 1
[.,[[.,[.,.]],[[.,.],[.,.]]]]
=> [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 5 - 1
[.,[[.,[.,.]],[[.,[.,.]],.]]]
=> [.,[[.,[[.,.],[.,.]]],[.,.]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ? = 3 - 1
[.,[[.,[.,.]],[[[.,.],.],.]]]
=> [.,[[.,[[[.,.],.],.]],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 5 - 1
[.,[[[.,.],.],[[.,[.,.]],.]]]
=> [.,[[[.,[[.,.],[.,.]]],.],.]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? = 5 - 1
[.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 5 - 1
[.,[[.,[.,[.,.]]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 5 - 1
[.,[[.,[[.,.],.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 5 - 1
[.,[[.,[[.,.],.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[[.,.],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 5 - 1
[.,[[[.,.],[.,.]],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[[[.,.],[.,.]],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? = 5 - 1
[.,[[[.,[.,.]],.],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],.],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 5 - 1
[.,[[[.,[.,.]],.],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],.],[.,.]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? = 5 - 1
[.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 5 - 1
[.,[[.,[.,[[.,.],.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[[.,.],.]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 5 - 1
[.,[[.,[[.,.],[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[[.,[.,.]],.]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? = 5 - 1
[.,[[.,[[.,[.,.]],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,3),(3,6),(5,6),(6,4)],7)
=> ? = 3 - 1
Description
The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset.
Matching statistic: St001032
Mp00018: Binary trees left border symmetryBinary trees
Mp00141: Binary trees pruning number to logarithmic heightDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
St001032: Dyck paths ⟶ ℤResult quality: 31% values known / values provided: 31%distinct values known / distinct values provided: 86%
Values
[.,.]
=> [.,.]
=> [1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[.,[.,.]]
=> [.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[[.,.],.]
=> [[.,.],.]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 4 - 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3 = 4 - 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 2 - 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 4 = 5 - 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 4 = 5 - 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 4 = 5 - 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 4 = 5 - 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 2 = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 4 = 5 - 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 4 = 5 - 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 4 = 5 - 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 4 = 5 - 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 2 = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 2 = 3 - 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 4 = 5 - 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [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
[.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> [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]
=> ? = 7 - 1
[.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [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]
=> ? = 7 - 1
[.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> [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]
=> ? = 5 - 1
[.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [.,[.,[.,[.,[[[.,.],.],.]]]]]
=> [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]
=> ? = 7 - 1
[.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [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]
=> ? = 7 - 1
[.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [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]
=> ? = 7 - 1
[.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],[.,.]]]]]
=> [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]
=> ? = 5 - 1
[.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [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]
=> ? = 7 - 1
[.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> [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]
=> ? = 5 - 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
=> [.,[.,[.,[[.,.],[[.,.],.]]]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 5 - 1
[.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [.,[.,[.,[[[.,.],[.,.]],.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 5 - 1
[.,[.,[.,[[[.,[.,.]],.],.]]]]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> [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]
=> ? = 5 - 1
[.,[.,[.,[[[[.,.],.],.],.]]]]
=> [.,[.,[.,[[[[.,.],.],.],.]]]]
=> [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]
=> ? = 7 - 1
[.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [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]
=> ? = 7 - 1
[.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [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]
=> ? = 7 - 1
[.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [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]
=> ? = 7 - 1
[.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 5 - 1
[.,[.,[[.,.],[[[.,.],.],.]]]]
=> [.,[.,[[.,[[[.,.],.],.]],.]]]
=> [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]
=> ? = 7 - 1
[.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [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]
=> ? = 5 - 1
[.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 5 - 1
[.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [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]
=> ? = 7 - 1
[.,[.,[[[.,.],.],[[.,.],.]]]]
=> [.,[.,[[[.,[[.,.],.]],.],.]]]
=> [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]
=> ? = 7 - 1
[.,[.,[[.,[.,[.,.]]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> [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]
=> ? = 5 - 1
[.,[.,[[.,[[.,.],.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> [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]
=> ? = 5 - 1
[.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [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]
=> ? = 5 - 1
[.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],[.,.]]]]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 5 - 1
[.,[.,[[[[.,.],.],.],[.,.]]]]
=> [.,[.,[[[[.,[.,.]],.],.],.]]]
=> [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]
=> ? = 7 - 1
[.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> [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]
=> ? = 5 - 1
[.,[.,[[.,[.,[[.,.],.]]],.]]]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 5 - 1
[.,[.,[[.,[[.,.],[.,.]]],.]]]
=> [.,[.,[[.,.],[[.,[.,.]],.]]]]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 5 - 1
[.,[.,[[.,[[.,[.,.]],.]],.]]]
=> [.,[.,[[.,.],[[.,.],[.,.]]]]]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 3 - 1
[.,[.,[[.,[[[.,.],.],.]],.]]]
=> [.,[.,[[.,.],[[[.,.],.],.]]]]
=> [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]
=> ? = 5 - 1
[.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [.,[.,[[[.,.],[.,[.,.]]],.]]]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 5 - 1
[.,[.,[[[.,.],[[.,.],.]],.]]]
=> [.,[.,[[[.,.],[[.,.],.]],.]]]
=> [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]
=> ? = 5 - 1
[.,[.,[[[.,[.,.]],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 - 1
[.,[.,[[[[.,.],.],[.,.]],.]]]
=> [.,[.,[[[[.,.],[.,.]],.],.]]]
=> [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]
=> ? = 5 - 1
[.,[.,[[[.,[.,[.,.]]],.],.]]]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> [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]
=> ? = 5 - 1
[.,[.,[[[.,[[.,.],.]],.],.]]]
=> [.,[.,[[[.,.],.],[[.,.],.]]]]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 5 - 1
[.,[.,[[[[.,.],[.,.]],.],.]]]
=> [.,[.,[[[[.,.],.],[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 5 - 1
[.,[.,[[[[.,[.,.]],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],[.,.]]]]
=> [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]
=> ? = 5 - 1
[.,[.,[[[[[.,.],.],.],.],.]]]
=> [.,[.,[[[[[.,.],.],.],.],.]]]
=> [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]
=> ? = 7 - 1
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 7 - 1
[.,[[.,.],[.,[.,[[.,.],.]]]]]
=> [.,[[.,[.,[.,[[.,.],.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 7 - 1
[.,[[.,.],[.,[[.,.],[.,.]]]]]
=> [.,[[.,[.,[[.,[.,.]],.]]],.]]
=> [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]
=> ? = 7 - 1
[.,[[.,.],[.,[[.,[.,.]],.]]]]
=> [.,[[.,[.,[[.,.],[.,.]]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 5 - 1
[.,[[.,.],[.,[[[.,.],.],.]]]]
=> [.,[[.,[.,[[[.,.],.],.]]],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 7 - 1
[.,[[.,.],[[.,.],[.,[.,.]]]]]
=> [.,[[.,[[.,[.,[.,.]]],.]],.]]
=> [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]
=> ? = 7 - 1
[.,[[.,.],[[.,.],[[.,.],.]]]]
=> [.,[[.,[[.,[[.,.],.]],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 7 - 1
[.,[[.,.],[[.,[.,.]],[.,.]]]]
=> [.,[[.,[[.,[.,.]],[.,.]]],.]]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
Description
The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. In other words, this is the number of valleys and peaks whose first step is in odd position, the initial position equal to 1. The generating function is given in [1].
Mp00018: Binary trees left border symmetryBinary trees
Mp00013: Binary trees to posetPosets
St001880: Posets ⟶ ℤResult quality: 20% values known / values provided: 20%distinct values known / distinct values provided: 71%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ? = 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? = 2
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? = 2
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 2
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(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),(1,2),(2,3)],4)
=> ? = 2
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1
[[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1
[[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3
[[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3
[[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[[[[.,[.,.]],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[.,[.,[.,[[.,.],.]]]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[.,[.,[[.,.],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[.,[.,[[.,[.,.]],.]]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4
[.,[.,[.,[[[.,.],.],.]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[.,[[.,.],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[.,[[.,.],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[.,[[.,[.,.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4
[.,[.,[[[.,.],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[.,[[.,[.,[.,.]]],.]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4
[.,[.,[[.,[[.,.],.]],.]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4
[.,[.,[[[.,.],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4
[.,[.,[[[.,[.,.]],.],.]]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4
[.,[.,[[[[.,.],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[[.,.],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[[.,.],[.,[[.,.],.]]]]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[[.,.],[[.,.],[.,.]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[[.,.],[[.,[.,.]],.]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4
[.,[[.,.],[[[.,.],.],.]]]
=> [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4
[.,[[.,[.,.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4
[.,[[[.,.],.],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[[[.,.],.],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4
[.,[[.,[[.,.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4
[.,[[[.,.],[.,.]],[.,.]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4
[.,[[[.,[.,.]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4
[.,[[[[.,.],.],.],[.,.]]]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4
[.,[[[[[.,.],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[.,.],[.,[.,[[.,.],.]]]]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[.,.],[.,[[.,.],[.,.]]]]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[.,.],[.,[[[.,.],.],.]]]
=> [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[.,.],[[.,.],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[[.,.],[[.,.],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Mp00018: Binary trees left border symmetryBinary trees
Mp00013: Binary trees to posetPosets
St001879: Posets ⟶ ℤResult quality: 20% values known / values provided: 20%distinct values known / distinct values provided: 71%
Values
[.,.]
=> [.,.]
=> ([],1)
=> ? = 1 - 1
[.,[.,.]]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? = 2 - 1
[[.,.],.]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? = 2 - 1
[.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[.,[[.,.],.]]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[.,.],[.,.]]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[[.,[.,.]],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1 - 1
[[[.,.],.],.]
=> [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[.,[.,[[.,.],.]]]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[.,[[.,.],[.,.]]]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[.,[[.,[.,.]],.]]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 2 - 1
[.,[[[.,.],.],.]]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[.,.],[[.,.],.]]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[[.,[.,[.,.]]],.]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[[.,[[.,.],.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[[[.,.],[.,.]],.]
=> [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 2 - 1
[[[.,[.,.]],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 - 1
[[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[.,[.,[[.,.],.]]]]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[.,[[.,.],[.,.]]]]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[.,[[.,[.,.]],.]]]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3 - 1
[.,[.,[[[.,.],.],.]]]
=> [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[[.,.],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[[.,.],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[[.,[.,.]],[.,.]]]
=> [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[[.,[.,[.,.]]],.]]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 - 1
[.,[[.,[[.,.],.]],.]]
=> [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,.],[.,[[.,.],.]]]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,.],[[.,.],[.,.]]]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,.],[[.,[.,.]],.]]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[[.,.],.],[[.,.],.]]
=> [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,[.,[.,.]]],[.,.]]
=> [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[[.,[[.,.],.]],[.,.]]
=> [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 - 1
[[[.,[.,.]],.],[.,.]]
=> [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 - 1
[[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[[.,[.,[.,[.,.]]]],.]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 - 1
[[.,[.,[[.,.],.]]],.]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 - 1
[[.,[[.,.],[.,.]]],.]
=> [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 - 1
[[.,[[.,[.,.]],.]],.]
=> [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 - 1
[[.,[[[.,.],.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 - 1
[[[.,.],[.,[.,.]]],.]
=> [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 - 1
[[[.,.],[[.,.],.]],.]
=> [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 - 1
[[[.,[.,.]],[.,.]],.]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 1 - 1
[[[[.,.],.],[.,.]],.]
=> [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 3 - 1
[[[.,[.,[.,.]]],.],.]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[[[.,[[.,.],.]],.],.]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 3 - 1
[[[[.,.],[.,.]],.],.]
=> [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 - 1
[[[[.,[.,.]],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 - 1
[[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[.,[.,[.,[[.,.],.]]]]]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[.,[.,[[.,.],[.,.]]]]]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[.,[.,[[.,[.,.]],.]]]]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4 - 1
[.,[.,[.,[[[.,.],.],.]]]]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[.,[[.,.],[.,[.,.]]]]]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[.,[[.,.],[[.,.],.]]]]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[.,[[.,[.,.]],[.,.]]]]
=> [.,[.,[[.,[.,.]],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4 - 1
[.,[.,[[[.,.],.],[.,.]]]]
=> [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[.,[[.,[.,[.,.]]],.]]]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4 - 1
[.,[.,[[.,[[.,.],.]],.]]]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4 - 1
[.,[.,[[[.,.],[.,.]],.]]]
=> [.,[.,[[[.,.],[.,.]],.]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4 - 1
[.,[.,[[[.,[.,.]],.],.]]]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4 - 1
[.,[.,[[[[.,.],.],.],.]]]
=> [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[[.,.],[.,[.,[.,.]]]]]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[[.,.],[.,[[.,.],.]]]]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[[.,.],[[.,.],[.,.]]]]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[[.,.],[[.,[.,.]],.]]]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? = 4 - 1
[.,[[.,.],[[[.,.],.],.]]]
=> [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[[.,[.,.]],[.,[.,.]]]]
=> [.,[[.,[.,[.,.]]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 - 1
[.,[[.,[.,.]],[[.,.],.]]]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 - 1
[.,[[[.,.],.],[.,[.,.]]]]
=> [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[[[.,.],.],[[.,.],.]]]
=> [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[[.,[.,[.,.]]],[.,.]]]
=> [.,[[.,[.,.]],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4 - 1
[.,[[.,[[.,.],.]],[.,.]]]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? = 4 - 1
[.,[[[.,.],[.,.]],[.,.]]]
=> [.,[[[.,[.,.]],[.,.]],.]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? = 4 - 1
[.,[[[.,[.,.]],.],[.,.]]]
=> [.,[[[.,[.,.]],.],[.,.]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 - 1
[.,[[[[.,.],.],.],[.,.]]]
=> [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[.,[[.,[.,[.,[.,.]]]],.]]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? = 4 - 1
[.,[[[[[.,.],.],.],.],.]]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[[.,.],[.,[.,[[.,.],.]]]]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[[.,.],[.,[[.,.],[.,.]]]]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[[.,.],[.,[[[.,.],.],.]]]
=> [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[[.,.],[[.,.],[.,[.,.]]]]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
[[.,.],[[.,.],[[.,.],.]]]
=> [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5 = 6 - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.