- FindStat

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Identifier

St001846: Lattices ⟶ ℤ

Values

([],1) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0

([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 0

([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => 0

([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => 0

([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7) => 5

([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7) => 5

([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7) => 5

([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => 5

([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 0

([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7) => 5

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 0

([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 5

([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => 5

([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 5

([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => 0

([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 0

([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => 5

([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 0

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7) => 0

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 5

([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => 0

([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => 5

([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7) => 5

([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7) => 0

([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 1

([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => 5

([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7) => 5

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7) => 2

([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7) => 5

([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7) => 0

([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7) => 0

([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 2

([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 5

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 3

([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => 0

([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7) => 2

([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 5

([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7) => 3

([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 5

([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7) => 5

([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7) => 1

([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 2

([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 0

([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 5

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7) => 0

([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 5

([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 0

([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 5

([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 3

([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 5

([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => 5

([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 5

([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 5

([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0

([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8) => 0

([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8) => 0

([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8) => 0

([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8) => 0

([(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4),(6,5)],8) => 6

([(0,6),(1,7),(2,7),(3,7),(4,7),(5,4),(6,1),(6,2),(6,3),(6,5)],8) => 6

([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4),(6,7)],8) => 6

([(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(5,4),(7,6)],8) => 6

([(0,6),(1,7),(2,7),(3,7),(4,7),(6,1),(6,2),(6,3),(6,4),(7,5)],8) => 6

([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2),(6,5)],8) => 0

([(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(6,5)],8) => 6

([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8) => 0

([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(7,6)],8) => 0

([(0,6),(2,7),(3,7),(4,7),(5,1),(6,2),(6,3),(6,4),(7,5)],8) => 6

([(0,5),(1,7),(2,6),(3,6),(4,1),(5,2),(5,3),(5,4),(6,7)],8) => 6

([(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2),(4,3),(5,7),(6,7)],8) => 6

([(0,5),(1,7),(2,7),(3,6),(4,3),(4,7),(5,1),(5,2),(5,4),(7,6)],8) => 6

([(0,5),(2,7),(3,6),(4,6),(5,2),(5,3),(5,4),(6,7),(7,1)],8) => 6

([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(5,4),(7,3)],8) => 6

([(0,6),(1,7),(2,7),(3,7),(4,3),(5,2),(6,1),(6,4),(6,5)],8) => 6

([(0,5),(1,7),(2,7),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4),(6,7)],8) => 6

([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8) => 0

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$F_{1} = 1$

$F_{2} = 1$

$F_{3} = q$

$F_{4} = 1 + q^{2}$

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

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

$F_{7} = 18 + 2\ q + 4\ q^{2} + 3\ q^{3} + 26\ q^{5}$

$F_{8} = 71 + 10\ q + 16\ q^{2} + 16\ q^{3} + 12\ q^{4} + 97\ q^{6}$

Description

The number of elements which do not have a complement in the lattice.
A complement of an element

$x$ in a lattice is an element

$y$ such that the meet of

$x$ and

$y$ is the bottom element and their join is the top element.

Code

def statistic(L):
    return sum(1 for e in L if not L.complements(e))

Created

Nov 14, 2022 at 21:02 by Martin Rubey

Updated

Nov 14, 2022 at 21:02 by Martin Rubey