- FindStat

Identifier

Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

[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) => 3

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

[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) => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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) => 4

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

[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) => 5

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

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

[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) => 4

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

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

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

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

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

[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) => 6

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

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

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

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

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

[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) => 3

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

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

[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) => 5

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

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

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

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

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

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

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

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

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

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

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

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

[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) => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 152 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + 3\ q^{2}$

$F_{4} = 1 + 3\ q + 7\ q^{2} + 11\ q^{3} + 2\ q^{4}$

$F_{5} = 1 + 4\ q + 12\ q^{2} + 28\ q^{3} + 48\ q^{4} + 22\ q^{5} + 5\ q^{6}$

Description

The number of cover relations in a poset.
Equivalently, this is also the number of edges in the Hasse diagram [1].

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

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