Identifier
-
Mp00018:
Binary trees
—left border symmetry⟶
Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001631: Posets ⟶ ℤ
Values
[.,.] => [.,.] => ([],1) => 0
[.,[.,.]] => [.,[.,.]] => ([(0,1)],2) => 1
[[.,.],.] => [[.,.],.] => ([(0,1)],2) => 1
[.,[.,[.,.]]] => [.,[.,[.,.]]] => ([(0,2),(2,1)],3) => 2
[.,[[.,.],.]] => [.,[[.,.],.]] => ([(0,2),(2,1)],3) => 2
[[.,.],[.,.]] => [[.,[.,.]],.] => ([(0,2),(2,1)],3) => 2
[[.,[.,.]],.] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => 0
[[[.,.],.],.] => [[[.,.],.],.] => ([(0,2),(2,1)],3) => 2
[.,[.,[.,[.,.]]]] => [.,[.,[.,[.,.]]]] => ([(0,3),(2,1),(3,2)],4) => 3
[.,[.,[[.,.],.]]] => [.,[.,[[.,.],.]]] => ([(0,3),(2,1),(3,2)],4) => 3
[.,[[.,.],[.,.]]] => [.,[[.,[.,.]],.]] => ([(0,3),(2,1),(3,2)],4) => 3
[.,[[.,[.,.]],.]] => [.,[[.,.],[.,.]]] => ([(0,3),(1,3),(3,2)],4) => 1
[.,[[[.,.],.],.]] => [.,[[[.,.],.],.]] => ([(0,3),(2,1),(3,2)],4) => 3
[[.,.],[.,[.,.]]] => [[.,[.,[.,.]]],.] => ([(0,3),(2,1),(3,2)],4) => 3
[[.,.],[[.,.],.]] => [[.,[[.,.],.]],.] => ([(0,3),(2,1),(3,2)],4) => 3
[[.,[.,.]],[.,.]] => [[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1
[[[.,.],.],[.,.]] => [[[.,[.,.]],.],.] => ([(0,3),(2,1),(3,2)],4) => 3
[[.,[.,[.,.]]],.] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 1
[[.,[[.,.],.]],.] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => 1
[[[.,.],[.,.]],.] => [[[.,.],[.,.]],.] => ([(0,3),(1,3),(3,2)],4) => 1
[[[.,[.,.]],.],.] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1
[[[[.,.],.],.],.] => [[[[.,.],.],.],.] => ([(0,3),(2,1),(3,2)],4) => 3
[.,[.,[.,[.,[.,.]]]]] => [.,[.,[.,[.,[.,.]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[.,[.,[.,[[.,.],.]]]] => [.,[.,[.,[[.,.],.]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[.,[.,[[.,.],[.,.]]]] => [.,[.,[[.,[.,.]],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[.,[.,[[.,[.,.]],.]]] => [.,[.,[[.,.],[.,.]]]] => ([(0,4),(1,4),(2,3),(4,2)],5) => 2
[.,[.,[[[.,.],.],.]]] => [.,[.,[[[.,.],.],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[.,[[.,.],[.,[.,.]]]] => [.,[[.,[.,[.,.]]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[.,[[.,.],[[.,.],.]]] => [.,[[.,[[.,.],.]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[.,[[.,[.,.]],[.,.]]] => [.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[.,[[[.,.],.],[.,.]]] => [.,[[[.,[.,.]],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[.,[[.,[.,[.,.]]],.]] => [.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[.,[[.,[[.,.],.]],.]] => [.,[[.,.],[[.,.],.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[.,[[[.,.],[.,.]],.]] => [.,[[[.,.],[.,.]],.]] => ([(0,4),(1,4),(2,3),(4,2)],5) => 2
[.,[[[.,[.,.]],.],.]] => [.,[[[.,.],.],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[.,[[[[.,.],.],.],.]] => [.,[[[[.,.],.],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[.,.],[.,[.,[.,.]]]] => [[.,[.,[.,[.,.]]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[.,.],[.,[[.,.],.]]] => [[.,[.,[[.,.],.]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[.,.],[[.,.],[.,.]]] => [[.,[[.,[.,.]],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[.,.],[[.,[.,.]],.]] => [[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(4,2)],5) => 2
[[.,.],[[[.,.],.],.]] => [[.,[[[.,.],.],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[.,[.,.]],[.,[.,.]]] => [[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[[.,[.,.]],[[.,.],.]] => [[.,[[.,.],.]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[[[.,.],.],[.,[.,.]]] => [[[.,[.,[.,.]]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[.,.],.],[[.,.],.]] => [[[.,[[.,.],.]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[.,[.,[.,.]]],[.,.]] => [[.,[.,.]],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2
[[.,[[.,.],.]],[.,.]] => [[.,[.,.]],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2
[[[.,.],[.,.]],[.,.]] => [[[.,[.,.]],[.,.]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[[[.,[.,.]],.],[.,.]] => [[[.,[.,.]],.],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[[[[.,.],.],.],[.,.]] => [[[[.,[.,.]],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[.,[.,[.,[.,.]]]],.] => [[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[[.,[.,[[.,.],.]]],.] => [[.,.],[.,[[.,.],.]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[[.,[[.,.],[.,.]]],.] => [[.,.],[[.,[.,.]],.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[[.,[[.,[.,.]],.]],.] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 0
[[.,[[[.,.],.],.]],.] => [[.,.],[[[.,.],.],.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[[[.,.],[.,[.,.]]],.] => [[[.,.],[.,[.,.]]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[[[.,.],[[.,.],.]],.] => [[[.,.],[[.,.],.]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[[[.,[.,.]],[.,.]],.] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 0
[[[[.,.],.],[.,.]],.] => [[[[.,.],[.,.]],.],.] => ([(0,4),(1,4),(2,3),(4,2)],5) => 2
[[[.,[.,[.,.]]],.],.] => [[[.,.],.],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2
[[[.,[[.,.],.]],.],.] => [[[.,.],.],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2
[[[[.,.],[.,.]],.],.] => [[[[.,.],.],[.,.]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[[[[.,[.,.]],.],.],.] => [[[[.,.],.],.],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[[[[[.,.],.],.],.],.] => [[[[[.,.],.],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[.,[.,[.,[.,[.,[.,.]]]]]] => [.,[.,[.,[.,[.,[.,.]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[.,[.,[.,[[.,.],.]]]]] => [.,[.,[.,[.,[[.,.],.]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[.,[.,[[.,.],[.,.]]]]] => [.,[.,[.,[[.,[.,.]],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[.,[.,[[.,[.,.]],.]]]] => [.,[.,[.,[[.,.],[.,.]]]]] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => 3
[.,[.,[.,[[[.,.],.],.]]]] => [.,[.,[.,[[[.,.],.],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[.,[[.,.],[.,[.,.]]]]] => [.,[.,[[.,[.,[.,.]]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[.,[[.,.],[[.,.],.]]]] => [.,[.,[[.,[[.,.],.]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[.,[[.,[.,.]],[.,.]]]] => [.,[.,[[.,[.,.]],[.,.]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 3
[.,[.,[[[.,.],.],[.,.]]]] => [.,[.,[[[.,[.,.]],.],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[.,[[.,[.,[.,.]]],.]]] => [.,[.,[[.,.],[.,[.,.]]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 3
[.,[.,[[.,[[.,.],.]],.]]] => [.,[.,[[.,.],[[.,.],.]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 3
[.,[.,[[[.,.],[.,.]],.]]] => [.,[.,[[[.,.],[.,.]],.]]] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => 3
[.,[.,[[[.,[.,.]],.],.]]] => [.,[.,[[[.,.],.],[.,.]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 3
[.,[.,[[[[.,.],.],.],.]]] => [.,[.,[[[[.,.],.],.],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[[.,.],[.,[.,[.,.]]]]] => [.,[[.,[.,[.,[.,.]]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[[.,.],[.,[[.,.],.]]]] => [.,[[.,[.,[[.,.],.]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[[.,.],[[.,.],[.,.]]]] => [.,[[.,[[.,[.,.]],.]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[[.,.],[[.,[.,.]],.]]] => [.,[[.,[[.,.],[.,.]]],.]] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => 3
[.,[[.,.],[[[.,.],.],.]]] => [.,[[.,[[[.,.],.],.]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[[.,[.,.]],[.,[.,.]]]] => [.,[[.,[.,[.,.]]],[.,.]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 3
[.,[[.,[.,.]],[[.,.],.]]] => [.,[[.,[[.,.],.]],[.,.]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 3
[.,[[[.,.],.],[.,[.,.]]]] => [.,[[[.,[.,[.,.]]],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[[[.,.],.],[[.,.],.]]] => [.,[[[.,[[.,.],.]],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[[.,[.,[.,.]]],[.,.]]] => [.,[[.,[.,.]],[.,[.,.]]]] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => 3
[.,[[.,[[.,.],.]],[.,.]]] => [.,[[.,[.,.]],[[.,.],.]]] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => 3
[.,[[[.,.],[.,.]],[.,.]]] => [.,[[[.,[.,.]],[.,.]],.]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 3
[.,[[[.,[.,.]],.],[.,.]]] => [.,[[[.,[.,.]],.],[.,.]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 3
[.,[[[[.,.],.],.],[.,.]]] => [.,[[[[.,[.,.]],.],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[.,[[.,[.,[.,[.,.]]]],.]] => [.,[[.,.],[.,[.,[.,.]]]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 3
[.,[[.,[.,[[.,.],.]]],.]] => [.,[[.,.],[.,[[.,.],.]]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 3
[.,[[.,[[.,.],[.,.]]],.]] => [.,[[.,.],[[.,[.,.]],.]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 3
[.,[[.,[[.,[.,.]],.]],.]] => [.,[[.,.],[[.,.],[.,.]]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => 1
[.,[[.,[[[.,.],.],.]],.]] => [.,[[.,.],[[[.,.],.],.]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 3
[.,[[[.,.],[.,[.,.]]],.]] => [.,[[[.,.],[.,[.,.]]],.]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 3
[.,[[[.,.],[[.,.],.]],.]] => [.,[[[.,.],[[.,.],.]],.]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 3
[.,[[[.,[.,.]],[.,.]],.]] => [.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => 1
[.,[[[[.,.],.],[.,.]],.]] => [.,[[[[.,.],[.,.]],.],.]] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => 3
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searching the database for statistics with the same generating function
Description
The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset.
Map
left border symmetry
Description
Return the tree where a symmetry has been applied recursively on all left borders. If a tree is made of three trees $T_1, T_2, T_3$ on its left border, it becomes $T_3, T_2, T_1$ where same symmetry has been applied to $T_1, T_2, T_3$.
Map
to poset
Description
Return the poset obtained by interpreting the tree as a Hasse diagram.
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