(2,2) – (2,0); (5,1) node\[fill,circle,scale=0.5pt\]; (5,1) node\[below\] [4123]{}; (5,4) node\[fill,circle,scale=0.5pt\]; (5,4) node\[above\] [45123]{}; (7,2.5) node\[fill,circle,scale=0.5pt\]; (7,1.7) node [456123]{}; (9,4) node\[fill,circle,scale=0.5pt\]; (9,4) node\[above\] [34512]{}; (9,1) node\[fill,circle,scale=0.5pt\]; (8.7,1) node\[below\] [2341]{}; (11,2.5) node\[fill,circle,scale=0.5pt\]; (11.75,2.5) node\[above\] [345612]{}; (11,0) node\[fill,circle,scale=0.5pt\]; (11,0) node\[below\] [23451]{}; (13,1) node\[fill,circle,scale=0.5pt\]; (13,1) node\[right\] [1234]{}; (7,2.5) – (5,1) – (5,4) – (7,2.5) – (9,4) – (9,1) – (7,2.5); (9,1) – (9,4) – (11,2.5) – (9,1) – (11,0) – (11,2.5) – (9,1); (11,0) – (13,1) – (11,2.5) – (11,0);

The Main Result {#sec:result}
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We use the map $f$ in Definition \[defn:map\] to calculate the Möbius function of $[\sigma,\pi]$ by calculating $\mu(A^{\sigma,\pi})$ and the effect on the Möbius function when applying $f$. First we compute $\mu(A^{\sigma,\pi})$. Given a set $A$ of posets the Möbius function of the union of $A$ can be calculated using the following inclusion-exclusion formula, which can be seen as a consequence of the inclusion-exclusion formula for the Euler characteristic and Equation : $$\label{eq:incexGen}\mu\left(\bigcup_{a\in A}a\right)=\sum_{\substack{S\subseteq A\\S\not=\emptyset}}(-1)^{|S|-1}\mu\left(\bigcap_{a\in S}a\right),$$ For more background on this see [@Nar74]. Applying Equation to $A^{\sigma,\pi}$ gives: $$\label{eq:incex}\mu(A^{\sigma,\pi})=\sum_{\substack{S\subseteq \widehat{E}^{\sigma,\pi}\\S\not=\emptyset}}(-1)^{|S|-1}\,\mu(\bigcap_{\eta\in S}P(\eta)^o).$$ To calculate this we need to know the Möbius function of the intersections $\cap_{\eta\in S} P(\eta)^o$. Note that when calculating the Möbius function of a poset we add 