Identifier
-
Mp00047:
Ordered trees
—to poset⟶
Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001031: Dyck paths ⟶ ℤ
Values
[[]] => ([(0,1)],2) => [2] => [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
[[],[],[]] => ([(0,3),(1,3),(2,3)],4) => [2,1,1] => [1,0,1,1,0,1,0,0] => 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),(1,3),(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
[[],[],[],[]] => ([(0,4),(1,4),(2,4),(3,4)],5) => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1
[[],[],[[]]] => ([(0,4),(1,4),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1
[[],[[]],[]] => ([(0,4),(1,4),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1
[[],[[],[]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 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),(1,4),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 1
[[[]],[[]]] => ([(0,3),(1,2),(2,4),(3,4)],5) => [3,2] => [1,0,1,1,1,0,0,0] => 1
[[[],[]],[]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 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),(1,4),(2,4),(4,3)],5) => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 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),(1,4),(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
[[],[],[],[],[]] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 1
[[],[],[],[[]]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[],[[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[],[[[]]]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[],[[]],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[]],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[[],[[],[]],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[[]]],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[],[[],[],[]]] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[],[[]]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[],[[[]],[]]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[],[[[],[]]]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[],[[[[]]]]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[[]],[],[],[]] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[[]],[],[[]]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[[[]],[[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 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] => 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
[[[],[]],[],[]] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[[[]]],[],[]] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 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] => 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
[[[],[],[]],[]] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[[],[[]]],[]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[[[]],[]],[]] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[[[],[]]],[]] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[[[[]]]],[]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[[],[],[],[]]] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6) => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[[],[],[[]]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[[],[[]],[]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[[],[[],[]]]] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[[],[[[]]]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[[[]],[],[]]] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 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),(1,4),(2,4),(4,5),(5,3)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[[[[]]],[]]] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[[[],[],[]]]] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[[[],[[]]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[[[[]],[]]]] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[[[[[],[]]]]] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6) => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 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
[[],[],[],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => [2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 1
[[],[],[],[],[[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 1
[[],[],[],[[]],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 1
[[],[],[],[[],[]]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 1
[[],[],[],[[[]]]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[],[[]],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 1
[[],[],[[]],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1
[[],[],[[],[]],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 1
[[],[],[[[]]],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[],[[],[],[]]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 1
[[],[],[[],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[],[[[]],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[],[[[],[]]]] => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[],[[[[]]]]] => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[],[[]],[],[],[]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 1
[[],[[]],[],[[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1
[[],[[]],[[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1
[[],[[]],[[],[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1
[[],[[]],[[[]]]] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7) => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 1
[[],[[],[]],[],[]] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 1
[[],[[[]]],[],[]] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[],[]],[[]]] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1
[[],[[[]]],[[]]] => ([(0,6),(1,3),(2,4),(3,5),(4,6),(5,6)],7) => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 1
[[],[[],[],[]],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(5,6)],7) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 1
[[],[[],[[]]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[[]],[]],[]] => ([(0,6),(1,6),(2,5),(3,4),(4,5),(5,6)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[[],[]]],[]] => ([(0,6),(1,6),(2,5),(3,5),(4,6),(5,4)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[[[]]]],[]] => ([(0,3),(1,6),(2,6),(3,5),(4,6),(5,4)],7) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[],[[],[],[],[]]] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => [3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 1
[[],[[],[],[[]]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[],[[]],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[],[[],[]]]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[],[[[]]]]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 1
[[],[[[]],[],[]]] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[[]],[[]]]] => ([(0,5),(1,4),(2,3),(3,6),(4,6),(6,5)],7) => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 1
[[],[[[],[]],[]]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 1
[[],[[[[]]],[]]] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7) => [5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 1
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searching the database for statistics with the same generating function
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
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 the Hasse diagram of a graph.
searching the database
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