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Statistic identifier: St001625
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Collection: Lattices
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Description: The Möbius invariant of a lattice.
The '''Möbius invariant''' of a lattice $L$ is the value of the Möbius function applied to least and greatest element, that is $\mu(L)=\mu_L(\hat{0},\hat{1})$, where $\hat{0}$ is the least element of $L$ and $\hat{1}$ is the greatest element of $L$.
For the definition of the Möbius function, see [[St000914]].
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References:
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Code:
def statistic(L):
return L.moebius_function(L.bottom(), L.top())
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Statistic values:
([],1) => 1
([(0,1)],2) => -1
([(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) => 2
([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 1
([(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) => 0
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 3
([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6) => 2
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6) => 1
([(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) => 1
([(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) => 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 0
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 0
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 0
([(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) => 0
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 4
([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7) => 3
([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7) => 2
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7) => 1
([(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) => 1
([(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) => 1
([(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) => 2
([(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2)],7) => 1
([(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) => 2
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7) => 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 0
([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7) => 1
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7) => 0
([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7) => 0
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7) => 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 0
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7) => 0
([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7) => 0
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7) => 0
([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7) => 0
([(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) => 1
([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 0
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 0
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 0
([(0,4),(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2)],7) => 1
([(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,2),(4,5),(5,6)],7) => 0
([(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) => 0
([(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) => 0
([(0,3),(0,4),(1,6),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6)],7) => 0
([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 0
([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 1
([(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) => 1
([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 0
([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 1
([(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) => 0
([(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) => 0
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 0
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 0
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Created: Oct 01, 2020 at 09:22 by Henri Mühle
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Last Updated: Feb 08, 2021 at 23:23 by Martin Rubey