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Identifier

Mp00065: Permutations —permutation poset⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions

Images

[1] => ([],1) => [1]

[1,2] => ([(0,1)],2) => [2]

[2,1] => ([],2) => [1,1]

[1,2,3] => ([(0,2),(2,1)],3) => [3]

[1,3,2] => ([(0,1),(0,2)],3) => [2,1]

[2,1,3] => ([(0,2),(1,2)],3) => [2,1]

[2,3,1] => ([(1,2)],3) => [2,1]

[3,1,2] => ([(1,2)],3) => [2,1]

[3,2,1] => ([],3) => [1,1,1]

[1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => [4]

[1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => [3,1]

[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => [3,1]

[1,3,4,2] => ([(0,2),(0,3),(3,1)],4) => [3,1]

[1,4,2,3] => ([(0,2),(0,3),(3,1)],4) => [3,1]

[1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => [2,1,1]

[2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => [3,1]

[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => [2,2]

[2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => [3,1]

[2,3,4,1] => ([(1,2),(2,3)],4) => [3,1]

[2,4,1,3] => ([(0,3),(1,2),(1,3)],4) => [2,2]

[2,4,3,1] => ([(1,2),(1,3)],4) => [2,1,1]

[3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => [3,1]

[3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => [2,2]

[3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => [2,1,1]

[3,2,4,1] => ([(1,3),(2,3)],4) => [2,1,1]

[3,4,1,2] => ([(0,3),(1,2)],4) => [2,2]

[3,4,2,1] => ([(2,3)],4) => [2,1,1]

[4,1,2,3] => ([(1,2),(2,3)],4) => [3,1]

[4,1,3,2] => ([(1,2),(1,3)],4) => [2,1,1]

[4,2,1,3] => ([(1,3),(2,3)],4) => [2,1,1]

[4,2,3,1] => ([(2,3)],4) => [2,1,1]

[4,3,1,2] => ([(2,3)],4) => [2,1,1]

[4,3,2,1] => ([],4) => [1,1,1,1]

[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => [5]

[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => [4,1]

[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [4,1]

[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5) => [4,1]

[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5) => [4,1]

[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => [3,1,1]

[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [4,1]

[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [3,2]

[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [4,1]

[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5) => [4,1]

[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2]

[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5) => [3,1,1]

[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [4,1]

[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2]

[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [3,1,1]

[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => [3,1,1]

[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5) => [3,2]

[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => [3,1,1]

[1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5) => [4,1]

[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5) => [3,1,1]

[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => [3,1,1]

[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => [3,1,1]

[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5) => [3,1,1]

[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => [2,1,1,1]

[2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => [4,1]

[2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => [3,2]

[2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [3,2]

[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [3,2]

[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [3,2]

[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1]

[2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1]

[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [3,2]

[2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => [4,1]

[2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => [4,1]

[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5) => [3,2]

[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5) => [3,1,1]

[2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2]

[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [3,2]

[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [3,1,1]

[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5) => [3,1,1]

[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5) => [3,2]

[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5) => [3,1,1]

[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => [3,2]

[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1]

[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5) => [3,1,1]

[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5) => [3,1,1]

[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1]

[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5) => [2,1,1,1]

[3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => [4,1]

[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [3,2]

[3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2]

[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5) => [3,2]

[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [3,2]

[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1]

[3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => [3,1,1]

[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1]

[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1]

[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5) => [3,1,1]

[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1]

[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1]

[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => [3,2]

[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5) => [3,2]

[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => [3,1,1]

[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5) => [3,1,1]

[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5) => [3,2]

[3,4,5,2,1] => ([(2,3),(3,4)],5) => [3,1,1]

[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5) => [3,2]

[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => [2,2,1]

>>> Load all 5994 entries. <<<

Map

permutation poset

Description

Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$

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.