Identifier
-
Mp00020:
Binary trees
—to Tamari-corresponding Dyck path⟶
Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001631: Posets ⟶ ℤ
Values
[.,.] => [1,0] => [.,.] => ([],1) => 0
[.,[.,.]] => [1,1,0,0] => [[.,.],.] => ([(0,1)],2) => 1
[[.,.],.] => [1,0,1,0] => [.,[.,.]] => ([(0,1)],2) => 1
[.,[.,[.,.]]] => [1,1,1,0,0,0] => [[.,.],[.,.]] => ([(0,2),(1,2)],3) => 0
[.,[[.,.],.]] => [1,1,0,1,0,0] => [[[.,.],.],.] => ([(0,2),(2,1)],3) => 2
[[.,.],[.,.]] => [1,0,1,1,0,0] => [.,[[.,.],.]] => ([(0,2),(2,1)],3) => 2
[[.,[.,.]],.] => [1,1,0,0,1,0] => [[.,[.,.]],.] => ([(0,2),(2,1)],3) => 2
[[[.,.],.],.] => [1,0,1,0,1,0] => [.,[.,[.,.]]] => ([(0,2),(2,1)],3) => 2
[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => [[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1
[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => [[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => 1
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => [[[.,.],[.,.]],.] => ([(0,3),(1,3),(3,2)],4) => 1
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => [[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => 1
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => [[[[.,.],.],.],.] => ([(0,3),(2,1),(3,2)],4) => 3
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => [.,[[.,.],[.,.]]] => ([(0,3),(1,3),(3,2)],4) => 1
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => [.,[[[.,.],.],.]] => ([(0,3),(2,1),(3,2)],4) => 3
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => [[.,[[.,.],.]],.] => ([(0,3),(2,1),(3,2)],4) => 3
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => [.,[.,[[.,.],.]]] => ([(0,3),(2,1),(3,2)],4) => 3
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => [[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => 1
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => [[[.,[.,.]],.],.] => ([(0,3),(2,1),(3,2)],4) => 3
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => [.,[[.,[.,.]],.]] => ([(0,3),(2,1),(3,2)],4) => 3
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => [[.,[.,[.,.]]],.] => ([(0,3),(2,1),(3,2)],4) => 3
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => [.,[.,[.,[.,.]]]] => ([(0,3),(2,1),(3,2)],4) => 3
[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => [[[.,.],[.,.]],[.,.]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 0
[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => [[[[.,.],.],.],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => [[.,[[.,.],.]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => [[[.,[.,.]],.],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => [[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => [[[[.,.],.],[.,.]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => [[[.,[.,.]],[.,.]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => [[.,.],[[.,.],[.,.]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => 0
[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => [[[[.,.],[.,.]],.],.] => ([(0,4),(1,4),(2,3),(4,2)],5) => 2
[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => [[[.,.],.],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2
[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => [[.,[.,.]],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2
[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => [[[.,.],[[.,.],.]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => [[.,.],[[[.,.],.],.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => [[[[[.,.],.],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => [.,[[[.,.],.],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => [.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => [.,[[[.,.],[.,.]],.]] => ([(0,4),(1,4),(2,3),(4,2)],5) => 2
[[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => [.,[[.,.],[[.,.],.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => [.,[[[[.,.],.],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => [[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(4,2)],5) => 2
[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => [[.,[[[.,.],.],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => [.,[.,[[.,.],[.,.]]]] => ([(0,4),(1,4),(2,3),(4,2)],5) => 2
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => [.,[.,[[[.,.],.],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => [[.,.],[.,[[.,.],.]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => [[[.,[[.,.],.]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => [.,[[.,[[.,.],.]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => [[.,[.,[[.,.],.]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => [.,[.,[.,[[.,.],.]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => [[[.,.],.],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2
[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => [[.,[.,.]],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 2
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => [[[.,.],[.,[.,.]]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => [[.,.],[[.,[.,.]],.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => [[[[.,[.,.]],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => [.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => 2
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => [.,[[[.,[.,.]],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => [[.,[[.,[.,.]],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => [.,[.,[[.,[.,.]],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => [[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 2
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => [[[.,[.,[.,.]]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => [.,[[.,[.,[.,.]]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => [[.,[.,[.,[.,.]]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => [.,[.,[.,[.,[.,.]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => [[[.,.],[.,.]],[[.,.],.]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => [[[.,.],[.,.]],[.,[.,.]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => [[[[.,.],[.,.]],.],[.,.]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => 1
[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => [[[[.,.],.],[.,.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 1
[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => [[[[[.,.],.],.],.],[.,.]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 3
[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => [[.,[[.,.],[.,.]]],[.,.]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => 1
[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => [[.,[[[.,.],.],.]],[.,.]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 3
[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => [[[.,[[.,.],.]],.],[.,.]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 3
[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => [[.,[.,[[.,.],.]]],[.,.]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 3
[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => [[[.,[.,.]],[.,.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 1
[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => [[[[.,[.,.]],.],.],[.,.]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 3
[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => [[.,[[.,[.,.]],.]],[.,.]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 3
[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => [[[.,[.,[.,.]]],.],[.,.]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 3
[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => [[.,[.,[.,[.,.]]]],[.,.]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 3
[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => [[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => 1
[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => [[[[[.,.],.],.],[.,.]],.] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 3
[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => [[[.,[[.,.],.]],[.,.]],.] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 3
[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => [[[[.,[.,.]],.],[.,.]],.] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 3
[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => [[[.,[.,[.,.]]],[.,.]],.] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 3
[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => [[.,.],[[[.,.],.],[.,.]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 1
[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => [[.,.],[[.,[.,.]],[.,.]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 1
[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => [[[[[.,.],.],[.,.]],.],.] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 3
[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => [[[[.,[.,.]],[.,.]],.],.] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 3
[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => [[[.,.],.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => [[.,[.,.]],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => 1
[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => [[[.,.],[[.,.],[.,.]]],.] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => 1
[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => [[.,.],[[[.,.],[.,.]],.]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => 1
[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => [[[[[.,.],[.,.]],.],.],.] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => 3
[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => [[[.,.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 1
[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => [[[[.,.],.],.],[[.,.],.]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 3
[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => [[.,[[.,.],.]],[[.,.],.]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 3
[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => [[[.,[.,.]],.],[[.,.],.]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 3
[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => [[.,[.,[.,.]]],[[.,.],.]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 3
[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => [[[[.,.],.],[[.,.],.]],.] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => 3
[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => [[[.,[.,.]],[[.,.],.]],.] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => 3
[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => [[.,.],[[.,.],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 1
[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => [[[[.,.],[[.,.],.]],.],.] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 3
>>> Load all 260 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
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
logarithmic height to pruning number
Description
Francon's map from Dyck paths to binary trees.
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].
This bijection sends the logarithmic height of the Dyck path, St000920The logarithmic height of a Dyck path., to the pruning number of the binary tree, St000396The register function (or Horton-Strahler number) of a binary tree.. The implementation is a literal translation of Knuth's [2].
Map
to Tamari-corresponding Dyck path
Description
Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:
The bijection is defined recursively as follows:
- a leaf is associated with an empty Dyck path,
- a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$.
Map
to poset
Description
Return the poset obtained by interpreting the tree as a Hasse diagram.
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!