- FindStat

Identifier

Mp00065: Permutations —permutation poset⟶ Posets
St001397: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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) => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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) => 2

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

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

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

[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) => 5

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

[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) => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

[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) => 5

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

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

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

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

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

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

[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) => 5

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

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

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

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

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

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

[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) => 4

[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) => 3

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

[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) => 5

[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) => 6

[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) => 5

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 3\ q + 5\ q^{2} + 6\ q^{3} + 5\ q^{4} + 3\ q^{5} + q^{6}$

$F_{5} = 1 + 4\ q + 9\ q^{2} + 15\ q^{3} + 20\ q^{4} + 22\ q^{5} + 20\ q^{6} + 15\ q^{7} + 9\ q^{8} + 4\ q^{9} + q^{10}$

$F_{6} = 1 + 5\ q + 14\ q^{2} + 29\ q^{3} + 49\ q^{4} + 71\ q^{5} + 90\ q^{6} + 101\ q^{7} + 101\ q^{8} + 90\ q^{9} + 71\ q^{10} + 49\ q^{11} + 29\ q^{12} + 14\ q^{13} + 5\ q^{14} + q^{15}$

Description

Number of pairs of incomparable elements in a finite poset.
For a finite poset

$(P,\leq)$ , this is the number of unordered pairs

$\{x,y\} \in \binom{P}{2}$ with

$x \not\leq y$ and

$y \not\leq x$ .

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

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