so the latter term of Equation is nonzero in this case. Furthermore, using Equation  we were able to compute the Möbius function of a rank 16 interval in 1 hour and a rank 17 interval in 6 hours. However, if $\sigma$ has a large number of occurrences in $\pi$ then using Equation can be quite slow. For example, if $\sigma=2413$ and $\pi= 2\, 4\, 6\, 8\, 10\, 1\, 3\, 5\, 7\, 9$ then there are 35 occurrences of $\sigma$ in $\pi$ and $\mu(\sigma,\pi)$ can be computed in 0.06 seconds using the recursive formula but takes 15.5 hours using Equation .

Poset Fibration {#sec:fibration}
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In this subsection we present a generalisation of the argument used to prove Theorem \[thm:main\]. We can view the pair $((\sigma,\pi),\{\widehat{E}^{\lambda,\pi}\}_{\lambda\in(\sigma,\pi)})$ as a poset fibration which makes $f$ the projection map and $A^{\sigma,\pi}$ the total space. In [@Bjo05] various theorems are presented which relate two posets $P$ and $Q$ linked by a poset fibration $f$ satisfying a certain condition, see Theorem 2.5 of [@Bjo05] for the most general form of this condition. However, our poset fibration does not always satisfy this condition, for example the condition is not true on the interval $[1,456123]$. We present a 