- FindStat

Identifier

Mp00125: Posets —dual poset⟶ Posets
St000846: Posets ⟶ ℤ

Values

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

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

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

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

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

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

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

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

([],4) => ([],4) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],5) => ([],5) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => ([],6) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 1 + 8\ q + 6\ q^{2} + q^{3}$

$F_{5} = 1 + 19\ q + 34\ q^{2} + 8\ q^{3} + q^{4}$

$F_{6} = 1 + 47\ q + 189\ q^{2} + 70\ q^{3} + 10\ q^{4} + q^{5}$

Description

The maximal number of elements covering an element of a poset.

Map

dual poset

Description

The dual of a poset.
The dual (or opposite) of a poset

$(\mathcal P,\leq)$ is the poset

$(\mathcal P^d,\leq_d)$ with

$x \leq_d y$ if

$y \leq x$ .