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Identifier

Mp00008: Binary trees —to complete tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000319: Integer partitions ⟶ ℤ (values match St000320The dinv adjustment of an integer partition.)

Values

[.,.] => [[],[]] => ([(0,2),(1,2)],3) => [2,1] => 1
[.,[.,.]] => [[],[[],[]]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => 2
[[.,.],.] => [[[],[]],[]] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => 2
[.,[.,[.,.]]] => [[],[[],[[],[]]]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => 3
[.,[[.,.],.]] => [[],[[[],[]],[]]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => 3
[[.,.],[.,.]] => [[[],[]],[[],[]]] => ([(0,5),(1,5),(2,4),(3,4),(4,6),(5,6)],7) => [3,2,1,1] => 2
[[.,[.,.]],.] => [[[],[[],[]]],[]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => 3
[[[.,.],.],.] => [[[[],[]],[]],[]] => ([(0,4),(1,4),(2,5),(3,6),(4,6),(6,5)],7) => [4,1,1,1] => 3
[.,[.,[.,[.,.]]]] => [[],[[],[[],[[],[]]]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 4
[.,[.,[[.,.],.]]] => [[],[[],[[[],[]],[]]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 4
[.,[[.,.],[.,.]]] => [[],[[[],[]],[[],[]]]] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9) => [4,2,1,1,1] => 3
[.,[[.,[.,.]],.]] => [[],[[[],[[],[]]],[]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 4
[.,[[[.,.],.],.]] => [[],[[[[],[]],[]],[]]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 4
[[.,.],[.,[.,.]]] => [[[],[]],[[],[[],[]]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => 3
[[.,.],[[.,.],.]] => [[[],[]],[[[],[]],[]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => 3
[[.,[.,.]],[.,.]] => [[[],[[],[]]],[[],[]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => 3
[[[.,.],.],[.,.]] => [[[[],[]],[]],[[],[]]] => ([(0,7),(1,5),(2,5),(3,6),(4,6),(5,8),(6,7),(7,8)],9) => [4,2,1,1,1] => 3
[[.,[.,[.,.]]],.] => [[[],[[],[[],[]]]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 4
[[.,[[.,.],.]],.] => [[[],[[[],[]],[]]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 4
[[[.,.],[.,.]],.] => [[[[],[]],[[],[]]],[]] => ([(0,7),(1,6),(2,6),(3,5),(4,5),(5,8),(6,8),(8,7)],9) => [4,2,1,1,1] => 3
[[[.,[.,.]],.],.] => [[[[],[[],[]]],[]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 4
[[[[.,.],.],.],.] => [[[[[],[]],[]],[]],[]] => ([(0,5),(1,5),(2,7),(3,8),(4,6),(5,8),(7,6),(8,7)],9) => [5,1,1,1,1] => 4

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$F_{1} = q$

$F_{2} = 2\ q^{2}$

$F_{3} = q^{2} + 4\ q^{3}$

$F_{4} = 6\ q^{3} + 8\ q^{4}$

Description

The spin of an integer partition.
The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$
The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross.
This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

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.

Map

to complete tree

Description

Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes.