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Statistic identifier: St000551
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Collection: Lattices
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Description: The number of left modular elements of a lattice.
A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.
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References: [1] Liu, S.-C., Sagan, B. Left-modular elements [[arXiv:math/0001055]]
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Code:
def is_left_modular(L, x):
for z in L:
for y in L.principal_order_ideal(z):
if L.meet(L.join(x,y),z) != L.join(L.meet(x,z),y):
return False
return True
def statistic(L):
return len([x for x in L if is_left_modular(L, x)])
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Statistic values:
([],1) => 1
([(0,1)],2) => 2
([(0,2),(2,1)],3) => 3
([(0,1),(0,2),(1,3),(2,3)],4) => 4
([(0,3),(2,1),(3,2)],4) => 4
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 5
([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
([(0,4),(2,3),(3,1),(4,2)],5) => 5
([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 6
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 4
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 5
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6) => 6
([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 5
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 6
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 5
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 6
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 6
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 2
([(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) => 6
([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 7
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 4
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => 5
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => 6
([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7) => 7
([(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) => 6
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7) => 7
([(0,3),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1),(5,4)],7) => 5
([(0,5),(1,6),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7) => 6
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,1),(4,2),(5,6)],7) => 6
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 7
([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => 3
([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 7
([(0,2),(0,3),(0,5),(1,6),(2,6),(3,6),(4,1),(5,4)],7) => 5
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 6
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7) => 7
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 5
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7) => 6
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 7
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => 6
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => 7
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7) => 7
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7) => 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 5
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => 7
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7) => 6
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7) => 7
([(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) => 2
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7) => 6
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 5
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 7
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 6
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => 2
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7) => 7
([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 6
([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7) => 5
([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 6
([(0,4),(0,5),(1,6),(2,6),(4,6),(5,1),(5,2),(6,3)],7) => 6
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7) => 5
([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 5
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 6
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 7
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7) => 5
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 6
([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 2
([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 6
([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 6
([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => 3
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 6
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 7
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Created: Jul 19, 2016 at 23:49 by Martin Rubey
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Last Updated: Jul 21, 2016 at 10:49 by Martin Rubey