Identifier
Values
[.,.] => [[],[]] => ([(0,2),(1,2)],3) => [2,1] => 1
[.,[.,.]] => [[],[[],[]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => 3
[[.,.],.] => [[[],[]],[]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => 3
[.,[.,[.,.]]] => [[],[[],[[],[]]]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => 6
[.,[[.,.],.]] => [[],[[[],[]],[]]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => 6
[[.,.],[.,.]] => [[[],[]],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => 4
[[.,[.,.]],.] => [[[],[[],[]]],[]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => 6
[[[.,.],.],.] => [[[[],[]],[]],[]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => 6
[.,[.,[.,[.,.]]]] => [[],[[],[[],[[],[]]]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 10
[.,[.,[[.,.],.]]] => [[],[[],[[[],[]],[]]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 10
[.,[[.,.],[.,.]]] => [[],[[[],[]],[[],[]]]] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9) => [4,2,1,1,1] => 7
[.,[[.,[.,.]],.]] => [[],[[[],[[],[]]],[]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 10
[.,[[[.,.],.],.]] => [[],[[[[],[]],[]],[]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 10
[[.,.],[.,[.,.]]] => [[[],[]],[[],[[],[]]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => 7
[[.,.],[[.,.],.]] => [[[],[]],[[[],[]],[]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => 7
[[.,[.,.]],[.,.]] => [[[],[[],[]]],[[],[]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => 7
[[[.,.],.],[.,.]] => [[[[],[]],[]],[[],[]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => 7
[[.,[.,[.,.]]],.] => [[[],[[],[[],[]]]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 10
[[.,[[.,.],.]],.] => [[[],[[[],[]],[]]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 10
[[[.,.],[.,.]],.] => [[[[],[]],[[],[]]],[]] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9) => [4,2,1,1,1] => 7
[[[.,[.,.]],.],.] => [[[[],[[],[]]],[]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 10
[[[[.,.],.],.],.] => [[[[[],[]],[]],[]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 10
[.,[.,[.,[.,[.,.]]]]] => [[],[[],[[],[[],[[],[]]]]]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[.,[.,[.,[[.,.],.]]]] => [[],[[],[[],[[[],[]],[]]]]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[.,[.,[[.,.],[.,.]]]] => [[],[[],[[[],[]],[[],[]]]]] => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11) => [5,2,1,1,1,1] => 11
[.,[.,[[.,[.,.]],.]]] => [[],[[],[[[],[[],[]]],[]]]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[.,[.,[[[.,.],.],.]]] => [[],[[],[[[[],[]],[]],[]]]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[.,[[.,.],[.,[.,.]]]] => [[],[[[],[]],[[],[[],[]]]]] => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11) => [5,2,1,1,1,1] => 11
[.,[[.,.],[[.,.],.]]] => [[],[[[],[]],[[[],[]],[]]]] => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11) => [5,2,1,1,1,1] => 11
[.,[[.,[.,.]],[.,.]]] => [[],[[[],[[],[]]],[[],[]]]] => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11) => [5,2,1,1,1,1] => 11
[.,[[[.,.],.],[.,.]]] => [[],[[[[],[]],[]],[[],[]]]] => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11) => [5,2,1,1,1,1] => 11
[.,[[.,[.,[.,.]]],.]] => [[],[[[],[[],[[],[]]]],[]]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[.,[[.,[[.,.],.]],.]] => [[],[[[],[[[],[]],[]]],[]]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[.,[[[.,.],[.,.]],.]] => [[],[[[[],[]],[[],[]]],[]]] => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11) => [5,2,1,1,1,1] => 11
[.,[[[.,[.,.]],.],.]] => [[],[[[[],[[],[]]],[]],[]]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[.,[[[[.,.],.],.],.]] => [[],[[[[[],[]],[]],[]],[]]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[[.,.],[.,[.,[.,.]]]] => [[[],[]],[[],[[],[[],[]]]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11) => [5,2,1,1,1,1] => 11
[[.,.],[.,[[.,.],.]]] => [[[],[]],[[],[[[],[]],[]]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11) => [5,2,1,1,1,1] => 11
[[.,.],[[.,.],[.,.]]] => [[[],[]],[[[],[]],[[],[]]]] => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11) => [4,2,2,1,1,1] => 8
[[.,.],[[.,[.,.]],.]] => [[[],[]],[[[],[[],[]]],[]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11) => [5,2,1,1,1,1] => 11
[[.,.],[[[.,.],.],.]] => [[[],[]],[[[[],[]],[]],[]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11) => [5,2,1,1,1,1] => 11
[[.,[.,.]],[.,[.,.]]] => [[[],[[],[]]],[[],[[],[]]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11) => [4,3,1,1,1,1] => 9
[[.,[.,.]],[[.,.],.]] => [[[],[[],[]]],[[[],[]],[]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11) => [4,3,1,1,1,1] => 9
[[[.,.],.],[.,[.,.]]] => [[[[],[]],[]],[[],[[],[]]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11) => [4,3,1,1,1,1] => 9
[[[.,.],.],[[.,.],.]] => [[[[],[]],[]],[[[],[]],[]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,8),(7,9),(8,10),(9,10)],11) => [4,3,1,1,1,1] => 9
[[.,[.,[.,.]]],[.,.]] => [[[],[[],[[],[]]]],[[],[]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11) => [5,2,1,1,1,1] => 11
[[.,[[.,.],.]],[.,.]] => [[[],[[[],[]],[]]],[[],[]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11) => [5,2,1,1,1,1] => 11
[[[.,.],[.,.]],[.,.]] => [[[[],[]],[[],[]]],[[],[]]] => ([(0,8),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,9),(8,9),(9,10)],11) => [4,2,2,1,1,1] => 8
[[[.,[.,.]],.],[.,.]] => [[[[],[[],[]]],[]],[[],[]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11) => [5,2,1,1,1,1] => 11
[[[[.,.],.],.],[.,.]] => [[[[[],[]],[]],[]],[[],[]]] => ([(0,9),(1,8),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,9),(9,10)],11) => [5,2,1,1,1,1] => 11
[[.,[.,[.,[.,.]]]],.] => [[[],[[],[[],[[],[]]]]],[]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[[.,[.,[[.,.],.]]],.] => [[[],[[],[[[],[]],[]]]],[]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[[.,[[.,.],[.,.]]],.] => [[[],[[[],[]],[[],[]]]],[]] => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11) => [5,2,1,1,1,1] => 11
[[.,[[.,[.,.]],.]],.] => [[[],[[[],[[],[]]],[]]],[]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[[.,[[[.,.],.],.]],.] => [[[],[[[[],[]],[]],[]]],[]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[[[.,.],[.,[.,.]]],.] => [[[[],[]],[[],[[],[]]]],[]] => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11) => [5,2,1,1,1,1] => 11
[[[.,.],[[.,.],.]],.] => [[[[],[]],[[[],[]],[]]],[]] => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11) => [5,2,1,1,1,1] => 11
[[[.,[.,.]],[.,.]],.] => [[[[],[[],[]]],[[],[]]],[]] => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11) => [5,2,1,1,1,1] => 11
[[[[.,.],.],[.,.]],.] => [[[[[],[]],[]],[[],[]]],[]] => ([(0,8),(1,9),(2,6),(3,6),(4,7),(5,7),(6,10),(7,8),(8,10),(10,9)],11) => [5,2,1,1,1,1] => 11
[[[.,[.,[.,.]]],.],.] => [[[[],[[],[[],[]]]],[]],[]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[[[.,[[.,.],.]],.],.] => [[[[],[[[],[]],[]]],[]],[]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[[[[.,.],[.,.]],.],.] => [[[[[],[]],[[],[]]],[]],[]] => ([(0,9),(1,8),(2,7),(3,7),(4,6),(5,6),(6,10),(7,10),(8,9),(10,8)],11) => [5,2,1,1,1,1] => 11
[[[[.,[.,.]],.],.],.] => [[[[[],[[],[]]],[]],[]],[]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
[[[[[.,.],.],.],.],.] => [[[[[[],[]],[]],[]],[]],[]] => ([(0,6),(1,6),(2,8),(3,9),(4,10),(5,7),(6,10),(8,9),(9,7),(10,8)],11) => [6,1,1,1,1,1] => 15
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Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Map
to complete tree
Description
Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes.
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.
Map
to poset
Description
Return the poset obtained by interpreting the tree as the Hasse diagram of a graph.