Identifier
-
Mp00013:
Binary trees
—to poset⟶
Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001431: Dyck paths ⟶ ℤ
Values
[.,[.,.]] => ([(0,1)],2) => [2] => [1,0,1,0] => 0
[[.,.],.] => ([(0,1)],2) => [2] => [1,0,1,0] => 0
[.,[.,[.,.]]] => ([(0,2),(2,1)],3) => [3] => [1,0,1,0,1,0] => 0
[.,[[.,.],.]] => ([(0,2),(2,1)],3) => [3] => [1,0,1,0,1,0] => 0
[[.,.],[.,.]] => ([(0,2),(1,2)],3) => [2,1] => [1,0,1,1,0,0] => 1
[[.,[.,.]],.] => ([(0,2),(2,1)],3) => [3] => [1,0,1,0,1,0] => 0
[[[.,.],.],.] => ([(0,2),(2,1)],3) => [3] => [1,0,1,0,1,0] => 0
[.,[.,[.,[.,.]]]] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,0,1,0,1,0,1,0] => 0
[.,[.,[[.,.],.]]] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,0,1,0,1,0,1,0] => 0
[.,[[.,.],[.,.]]] => ([(0,3),(1,3),(3,2)],4) => [3,1] => [1,0,1,0,1,1,0,0] => 1
[.,[[.,[.,.]],.]] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,0,1,0,1,0,1,0] => 0
[.,[[[.,.],.],.]] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,0,1,0,1,0,1,0] => 0
[[.,.],[.,[.,.]]] => ([(0,3),(1,2),(2,3)],4) => [3,1] => [1,0,1,0,1,1,0,0] => 1
[[.,.],[[.,.],.]] => ([(0,3),(1,2),(2,3)],4) => [3,1] => [1,0,1,0,1,1,0,0] => 1
[[.,[.,.]],[.,.]] => ([(0,3),(1,2),(2,3)],4) => [3,1] => [1,0,1,0,1,1,0,0] => 1
[[[.,.],.],[.,.]] => ([(0,3),(1,2),(2,3)],4) => [3,1] => [1,0,1,0,1,1,0,0] => 1
[[.,[.,[.,.]]],.] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,0,1,0,1,0,1,0] => 0
[[.,[[.,.],.]],.] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,0,1,0,1,0,1,0] => 0
[[[.,.],[.,.]],.] => ([(0,3),(1,3),(3,2)],4) => [3,1] => [1,0,1,0,1,1,0,0] => 1
[[[.,[.,.]],.],.] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,0,1,0,1,0,1,0] => 0
[[[[.,.],.],.],.] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,0,1,0,1,0,1,0] => 0
[.,[.,[.,[.,[.,.]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[.,[.,[.,[[.,.],.]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[.,[.,[[.,.],[.,.]]]] => ([(0,4),(1,4),(2,3),(4,2)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[.,[.,[[.,[.,.]],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[.,[.,[[[.,.],.],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[.,[[.,.],[.,[.,.]]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[.,[[.,.],[[.,.],.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[.,[[.,[.,.]],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[.,[[[.,.],.],[.,.]]] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[.,[[.,[.,[.,.]]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[.,[[.,[[.,.],.]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[.,[[[.,.],[.,.]],.]] => ([(0,4),(1,4),(2,3),(4,2)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[.,[[[.,[.,.]],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[.,[[[[.,.],.],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[.,.],[.,[.,[.,.]]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[.,.],[.,[[.,.],.]]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[.,.],[[.,.],[.,.]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
[[.,.],[[.,[.,.]],.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[.,.],[[[.,.],.],.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[.,[.,.]],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => 1
[[.,[.,.]],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => 1
[[[.,.],.],[.,[.,.]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => 1
[[[.,.],.],[[.,.],.]] => ([(0,3),(1,2),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => 1
[[.,[.,[.,.]]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[.,[[.,.],.]],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[[.,.],[.,.]],[.,.]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 2
[[[.,[.,.]],.],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[[[.,.],.],.],[.,.]] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[.,[.,[.,[.,.]]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[.,[.,[[.,.],.]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[.,[[.,.],[.,.]]],.] => ([(0,4),(1,4),(2,3),(4,2)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[.,[[.,[.,.]],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[.,[[[.,.],.],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[[.,.],[.,[.,.]]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[[.,.],[[.,.],.]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[[.,[.,.]],[.,.]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[[[.,.],.],[.,.]],.] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[[.,[.,[.,.]]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[[.,[[.,.],.]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[[[.,.],[.,.]],.],.] => ([(0,4),(1,4),(2,3),(4,2)],5) => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[[[[.,[.,.]],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[[[[[.,.],.],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,0,1,0,1,0,1,0,1,0] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[.,[.,[.,[[.,.],.]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[.,[.,[[.,[.,.]],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[.,[.,[[[.,.],.],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[.,[[.,[.,[.,.]]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[.,[[.,[[.,.],.]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[.,[[[.,[.,.]],.],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[.,[[[[.,.],.],.],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[[.,[.,.]],[.,[.,.]]]] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[.,[[.,[.,.]],[[.,.],.]]] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[.,[[[.,.],.],[.,[.,.]]]] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[.,[[[.,.],.],[[.,.],.]]] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[[.,[.,[[.,.],.]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[[.,[[.,[.,.]],.]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[[.,[[[.,.],.],.]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[[[.,[.,[.,.]]],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[[[.,[[.,.],.]],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[[[[.,[.,.]],.],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[.,[[[[[.,.],.],.],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[[.,[.,.]],[.,[.,[.,.]]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[.,[.,.]],[.,[[.,.],.]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[.,[.,.]],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[[.,[.,.]],[[.,[.,.]],.]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[.,[.,.]],[[[.,.],.],.]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[[.,.],.],[.,[.,[.,.]]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[[.,.],.],[.,[[.,.],.]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[[.,.],.],[[.,.],[.,.]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[[[.,.],.],[[.,[.,.]],.]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[[.,.],.],[[[.,.],.],.]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[.,[.,[.,.]]],[.,[.,.]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[.,[.,[.,.]]],[[.,.],.]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[.,[[.,.],.]],[.,[.,.]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[.,[[.,.],.]],[[.,.],.]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[[.,.],[.,.]],[.,[.,.]]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[[[.,.],[.,.]],[[.,.],.]] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[[[.,[.,.]],.],[.,[.,.]]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
[[[.,[.,.]],.],[[.,.],.]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => [4,2] => [1,0,1,0,1,1,1,0,0,0] => 1
>>> Load all 139 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
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
Greene-Kleitman invariant
Description
The Greene-Kleitman invariant of a poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.
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!