- FindStat

Identifier

St001890: Posets ⟶ ℤ

Values

=>
                            Cc0014;cc-rep
                            
                            
                            

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

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Description

The maximum magnitude of the Möbius function of a poset.
The Möbius function of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.

References

[1] wikipedia:Incidence_algebra#Special_elements

Code

def statistic(P):
    return  max(abs(P.moebius_function(x,y)) for x in P for y in P)

Created

Mar 13, 2023 at 22:17 by Harry Richman

Updated

Mar 13, 2023 at 22:17 by Harry Richman