- FindStat

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Identifier

St000914: Posets ⟶ ℤ

Values

([],2) => 2

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

([],3) => 3

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

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

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

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

([],4) => 4

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

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

([(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) => 2

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

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

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

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

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

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

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

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

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

([],5) => 5

([(3,4)],5) => 4

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

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

([(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) => 1

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

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

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

([(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) => 3

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

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

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

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

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

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

([(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) => 2

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

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

([(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) => 3

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

([(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) => 0

([(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) => 2

([(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) => 0

([(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) => 0

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

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

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

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

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

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

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

([(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

([],6) => 6

([(4,5)],6) => 5

([(3,4),(3,5)],6) => 4

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$ \sum_{x\leq y} \mu(x,y). $$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.

Code

def statistic(P):
    return sum(P.moebius_function(x, y) for x in P for y in P if P.le(x,y))

Created

Jul 20, 2017 at 23:57 by Martin Rubey

Updated

Nov 13, 2022 at 10:44 by Martin Rubey