Identifier
-
Mp00013:
Binary trees
—to poset⟶
Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ
Values
[.,.] => ([],1) => [1] => [1,0,1,0] => 1
[.,[.,.]] => ([(0,1)],2) => [2] => [1,1,0,0,1,0] => 1
[[.,.],.] => ([(0,1)],2) => [2] => [1,1,0,0,1,0] => 1
[.,[.,[.,.]]] => ([(0,2),(2,1)],3) => [3] => [1,1,1,0,0,0,1,0] => 1
[.,[[.,.],.]] => ([(0,2),(2,1)],3) => [3] => [1,1,1,0,0,0,1,0] => 1
[[.,[.,.]],.] => ([(0,2),(2,1)],3) => [3] => [1,1,1,0,0,0,1,0] => 1
[[[.,.],.],.] => ([(0,2),(2,1)],3) => [3] => [1,1,1,0,0,0,1,0] => 1
[.,[.,[.,[.,.]]]] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1
[.,[.,[[.,.],.]]] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1
[.,[[.,[.,.]],.]] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1
[.,[[[.,.],.],.]] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1
[[.,[.,[.,.]]],.] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1
[[.,[[.,.],.]],.] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1
[[[.,[.,.]],.],.] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1
[[[[.,.],.],.],.] => ([(0,3),(2,1),(3,2)],4) => [4] => [1,1,1,1,0,0,0,0,1,0] => 1
[.,[.,[.,[.,[.,.]]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[.,[.,[[.,.],.]]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[.,[[.,[.,.]],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[.,[[[.,.],.],.]]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[[.,[.,[.,.]]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[[.,[[.,.],.]],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[[[.,[.,.]],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[[[[.,.],.],.],.]] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[.,.],[[.,.],[.,.]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => 3
[[[.,.],[.,.]],[.,.]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,1,0,0,1,0] => 3
[[.,[.,[.,[.,.]]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[.,[.,[[.,.],.]]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[.,[[.,[.,.]],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[.,[[[.,.],.],.]],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[[.,[.,[.,.]]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[[.,[[.,.],.]],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[[[.,[.,.]],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[[[[.,.],.],.],.],.] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[.,[.,[.,[.,[.,[.,.]]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[.,[.,[[.,[.,.]],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[.,[.,[[[.,.],.],.]]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[.,[[.,[[.,.],.]],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[.,[[[.,[.,.]],.],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[.,[[[[.,.],.],.],.]]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[[.,.],[[.,.],[.,.]]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[.,[[[.,.],[.,.]],[.,.]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[.,[[.,[.,[.,[.,.]]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[[.,[.,[[.,.],.]]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[[.,[[.,[.,.]],.]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[[.,[[[.,.],.],.]],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[[[.,[.,[.,.]]],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[[[.,[[.,.],.]],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[[[[.,[.,.]],.],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[[[[[.,.],.],.],.],.]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[.,.],[.,[[.,.],[.,.]]]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[.,.],[[.,.],[.,[.,.]]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[.,.],[[.,.],[[.,.],.]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[.,.],[[.,[.,.]],[.,.]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[.,.],[[[.,.],.],[.,.]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[.,.],[[[.,.],[.,.]],.]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[.,[[.,.],[.,.]]],[.,.]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[[.,.],[.,[.,.]]],[.,.]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[[.,.],[[.,.],.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[[.,[.,.]],[.,.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[[[.,.],.],[.,.]],[.,.]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[[[.,.],[.,.]],.],[.,.]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[.,[.,[.,[.,[.,.]]]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[.,[.,[.,[[.,.],.]]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[.,[.,[[.,[.,.]],.]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[.,[.,[[[.,.],.],.]]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[.,[[.,[.,[.,.]]],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[.,[[.,[[.,.],.]],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[.,[[[.,[.,.]],.],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[.,[[[[.,.],.],.],.]],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[[.,.],[[.,.],[.,.]]],.] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[[[.,.],[.,.]],[.,.]],.] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [4,1,1] => [1,1,0,1,1,0,0,0,1,0] => 5
[[[.,[.,[.,[.,.]]]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[[.,[.,[[.,.],.]]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[[.,[[.,[.,.]],.]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[[.,[[[.,.],.],.]],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[[[.,[.,[.,.]]],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[[[.,[[.,.],.]],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[[[[.,[.,.]],.],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[[[[[.,.],.],.],.],.],.] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => [6] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[.,[.,[[.,.],[[.,.],[.,.]]]]] => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[.,[[[.,.],[.,.]],[.,.]]]] => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[.,.],[.,[[.,.],[.,.]]]]] => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[.,.],[[.,.],[.,[.,.]]]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[.,.],[[.,.],[[.,.],.]]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[.,.],[[.,[.,.]],[.,.]]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[.,.],[[[.,.],.],[.,.]]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[.,.],[[[.,.],[.,.]],.]]] => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[.,[[.,.],[.,.]]],[.,.]]] => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[[.,.],[.,[.,.]]],[.,.]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[[.,.],[[.,.],.]],[.,.]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[[.,[.,.]],[.,.]],[.,.]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[[[.,.],.],[.,.]],[.,.]]] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[[[.,.],[.,.]],.],[.,.]]] => ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[[.,.],[[.,.],[.,.]]],.]] => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[.,[[[[.,.],[.,.]],[.,.]],.]] => ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[[.,.],[.,[.,[[.,.],[.,.]]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(6,3)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[[.,.],[.,[[.,.],[.,[.,.]]]]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[[.,.],[.,[[.,.],[[.,.],.]]]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
[[.,.],[.,[[.,[.,.]],[.,.]]]] => ([(0,6),(1,5),(2,3),(3,6),(4,5),(6,4)],7) => [5,1,1] => [1,1,1,0,1,1,0,0,0,0,1,0] => 7
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Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
to poset
Description
Return the poset obtained by interpreting the tree as a Hasse diagram.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.
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.
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