if we can represent our problem as a lattice for which we already know the Möbius function.

Consider the interval $[\lambda^m,\lambda^n]$, for some indecomposable permutation $\lambda$, the embeddings of $\lambda^m$ in $\lambda^n$ can be considered as subsets of $[n]:=\{1,\ldots,n\}$ of size $m$. So we can regard our problem as that of computing the Möbius function of a sublattice of the Boolean lattice:

The *Boolean lattice* $B_n$ is the poset of subsets of $[n]$ with the partial order given by inclusion.

Define the *truncated Boolean lattice* $B_{n}^{\ge k}$ as the subposet of $B_n$ where all elements $a\in B_n$ such that $|a|< k$ are retracted to a single point $\hat{0}$. Similarly, define $B_{n}^{\le k}$ as the subposet of $B_n$ where all elements $a\in B_n$ such that $|a|> k$ are retracted to a single point $\hat{1}$.

(0,0) – (-3,1) – (-2,2) – (0,3); (0,0) – (-1.75,1) – (-2,2) – (0,3); (0,0) – (0.5,1) – (-0.5,2) – (0,3); (0,0) – (-0.5,1) – (-2,2) – (0,3); (0,0) – (1.75,1) – (2,2) – (0,3); (0,0) – (3,1) – (0.5,2) – (0,3); (3,1) – (2,2); (1.75,1) – (-0.5,2); (-0.5,1) – (2,2); (0.5,1) – (0.5,2); (-1.75,1) – (0.5,2); (-3,1) – (-0.5,2); (0,3) node\[above\] [$1234$]{}; (0,3) node\[fill,circle,scale=0.5pt\]; (-2,2.1) node\[left\] 