Values
=>
Cc0029;cc-rep
([],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)=>5
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)=>4
([(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)=>4
([(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)=>4
([(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)=>6
([(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)=>4
([(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)=>6
([(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)=>4
([(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)=>4
([(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)=>5
([(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)=>4
([(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)=>6
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)=>5
([(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)=>5
([(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)=>6
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,2),(5,1)],7)=>5
([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)=>5
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)=>6
([(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)=>4
([(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)=>5
([(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)=>5
([(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)=>6
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)=>4
([(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)=>4
([(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)=>5
([(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)=>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)=>7
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Description
The number of 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 $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
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 $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
References
[1] Liu, S.-C., Sagan, B. Left-modular elements arXiv:math/0001055
Code
def statistic(L):
return len([x for x in L if LatticePoset(L).is_modular_element(x)])
Created
Jul 19, 2016 at 23:41 by Martin Rubey
Updated
Sep 27, 2020 at 19:19 by Martin Rubey
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