- FindStat

Identifier

Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

[3,2,1] => ([],3) => 6

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

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

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

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

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

[1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 6

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

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

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

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

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

[2,4,3,1] => ([(1,2),(1,3)],4) => 8

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

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

[3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 6

[3,2,4,1] => ([(1,3),(2,3)],4) => 8

[3,4,1,2] => ([(0,3),(1,2)],4) => 6

[3,4,2,1] => ([(2,3)],4) => 12

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

[4,1,3,2] => ([(1,2),(1,3)],4) => 8

[4,2,1,3] => ([(1,3),(2,3)],4) => 8

[4,2,3,1] => ([(2,3)],4) => 12

[4,3,1,2] => ([(2,3)],4) => 12

[4,3,2,1] => ([],4) => 24

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

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

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

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

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

[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => 6

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

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

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

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

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

[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5) => 8

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

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

[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 6

[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 8

[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5) => 6

[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 12

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

[1,5,2,4,3] => ([(0,3),(0,4),(4,1),(4,2)],5) => 8

[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 8

[1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 12

[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(4,1)],5) => 12

[1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 24

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

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

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

[2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 6

[2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 6

[2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 12

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

[2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 6

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

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

[2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5) => 7

[2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5) => 10

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

[2,4,1,5,3] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 8

[2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 8

[2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5) => 10

[2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5) => 9

[2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5) => 15

[2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5) => 7

[2,5,1,4,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 14

[2,5,3,1,4] => ([(0,4),(1,2),(1,3),(3,4)],5) => 11

[2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5) => 15

[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(1,4)],5) => 18

[2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5) => 30

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

[3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 6

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

[3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5) => 7

[3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 8

[3,1,5,4,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 14

[3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => 6

[3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 12

[3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 8

[3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5) => 10

[3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 14

[3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5) => 20

[3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => 6

[3,4,1,5,2] => ([(0,3),(1,2),(1,4),(3,4)],5) => 9

[3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 12

[3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5) => 15

[3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5) => 10

[3,4,5,2,1] => ([(2,3),(3,4)],5) => 20

[3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5) => 9

[3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5) => 16

[3,5,2,1,4] => ([(0,4),(1,4),(2,3),(2,4)],5) => 18

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

$F_{3} = q + 2\ q^{2} + 2\ q^{3} + q^{6}$

$F_{4} = q + 3\ q^{2} + 4\ q^{3} + 3\ q^{4} + 2\ q^{5} + 3\ q^{6} + 4\ q^{8} + 3\ q^{12} + q^{24}$

$F_{5} = q + 4\ q^{2} + 6\ q^{3} + 7\ q^{4} + 6\ q^{5} + 9\ q^{6} + 4\ q^{7} + 10\ q^{8} + 4\ q^{9} + 8\ q^{10} + 2\ q^{11} + 8\ q^{12} + 4\ q^{14} + 8\ q^{15} + 2\ q^{16} + 4\ q^{18} + 9\ q^{20} + 2\ q^{24} + 4\ q^{25} + 7\ q^{30} + 6\ q^{40} + 4\ q^{60} + q^{120}$

$F_{6} = q + 5\ q^{2} + 8\ q^{3} + 12\ q^{4} + 10\ q^{5} + 21\ q^{6} + 8\ q^{7} + 21\ q^{8} + 16\ q^{9} + 22\ q^{10} + 8\ q^{11} + 31\ q^{12} + 4\ q^{13} + 18\ q^{14} + 22\ q^{15} + 16\ q^{16} + 33\ q^{18} + 8\ q^{19} + 23\ q^{20} + 4\ q^{21} + 4\ q^{22} + 4\ q^{23} + 35\ q^{24} + 8\ q^{25} + 10\ q^{26} + 8\ q^{27} + 12\ q^{28} + 37\ q^{30} + 8\ q^{32} + 8\ q^{33} + 4\ q^{35} + 19\ q^{36} + 4\ q^{37} + 6\ q^{38} + 22\ q^{40} + 16\ q^{42} + q^{44} + 8\ q^{45} + 26\ q^{48} + 8\ q^{52} + 12\ q^{54} + 2\ q^{57} + 27\ q^{60} + 2\ q^{61} + 4\ q^{64} + 6\ q^{66} + 4\ q^{70} + 16\ q^{72} + 4\ q^{75} + 2\ q^{76} + 4\ q^{80} + 12\ q^{84} + 17\ q^{90} + 4\ q^{96} + 8\ q^{108} + 21\ q^{120} + 4\ q^{144} + 6\ q^{150} + 12\ q^{180} + 8\ q^{240} + 5\ q^{360} + q^{720}$

Description

The number of linear extensions of a poset.

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

$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$