- FindStat

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Identifier

St001677: Lattices ⟶ ℤ

Values

([],1) => 0

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

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

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

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

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

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

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

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

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

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 11

([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 10

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

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

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

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

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

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

([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 9

([(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) => 8

([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 9

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

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

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

([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 26

([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 25

([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => 22

([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => 15

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

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

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

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

([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 15

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

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 15

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

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

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

([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => 22

([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 15

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

([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 24

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7) => 19

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

([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => 23

([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => 16

([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7) => 22

([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7) => 23

([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 20

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

([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7) => 18

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7) => 20

([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7) => 20

([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7) => 24

([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7) => 21

([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 14

([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 12

([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 12

([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => 21

([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7) => 14

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

([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7) => 14

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

([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7) => 14

([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7) => 18

([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 18

([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 15

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

([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7) => 21

([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 12

([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 21

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

([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 14

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

([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => 18

([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 14

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

([(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) => 57

([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1)],8) => 56

([(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2)],8) => 53

([(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3)],8) => 46

([(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,1),(6,2),(6,3),(6,4)],8) => 31

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

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

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

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

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

([(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,3),(6,1),(6,2),(6,5)],8) => 31

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

([(0,4),(0,5),(1,7),(2,6),(3,6),(4,7),(5,1),(5,2),(5,3),(6,7)],8) => 31

([(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,1),(5,2),(5,3),(7,6)],8) => 31

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

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

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

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

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

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

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

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

([(0,3),(0,4),(0,6),(1,7),(2,7),(3,7),(4,7),(5,2),(6,1),(6,5)],8) => 46

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$F_{1} = 1$

$F_{2} = 1$

$F_{3} = 1$

$F_{4} = 1 + q$

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

$F_{6} = 5 + q^{4} + 2\ q^{6} + 2\ q^{7} + q^{8} + 2\ q^{9} + q^{10} + q^{11}$

$F_{7} = 15 + 2\ q^{8} + 3\ q^{12} + 6\ q^{14} + 5\ q^{15} + q^{16} + 4\ q^{18} + q^{19} + 3\ q^{20} + 4\ q^{21} + 3\ q^{22} + 2\ q^{23} + 2\ q^{24} + q^{25} + q^{26}$

$F_{8} = 53 + 5\ q^{16} + 7\ q^{24} + 12\ q^{28} + 21\ q^{30} + 15\ q^{31} + 2\ q^{32} + 7\ q^{36} + 2\ q^{38} + 2\ q^{39} + 4\ q^{40} + 16\ q^{42} + 3\ q^{43} + 10\ q^{44} + 12\ q^{45} + 8\ q^{46} + q^{47} + 6\ q^{48} + 8\ q^{49} + 4\ q^{50} + 7\ q^{51} + 5\ q^{52} + 5\ q^{53} + 3\ q^{54} + 2\ q^{55} + q^{56} + q^{57}$

Description

The number of non-degenerate subsets of a lattice whose meet is the bottom element.
A subset whose meet is the bottom element is non-degenerate, if it neither contains the bottom, nor the top element of the lattice.

Code

def statistic(L):
    r = 0
    b = L.bottom()
    t = L.top()
    for s in subsets(L):
        if s and (b not in s) and (t not in s) and L.meet(s) == b:
            r += 1
    return r

Created

Feb 08, 2021 at 13:56 by Martin Rubey

Updated

Feb 08, 2021 at 13:56 by Martin Rubey