*component* of $\pi$, is indecomposable. Consider a permutation $\pi$, with a sequence of $\ell\ge2$ equal consecutive components, and $\lambda\le\pi$ obtained from $\pi$ by removing $k$ of the components from this sequence, where $\ell>k\ge1$. The interval $[\lambda,\pi]$ is disconnected, which follows from results in [@McSt13], specifically Lemma 4.2 and Theorem 5.6. These intervals are the cause of the extra conditions in the formulas for the Möbius function that appear in [@BJJS11] and [@McSt13].

The simplest permutations with equal consecutive components are the following:

Given an indecomposable permutation $\lambda$ let $$\lambda^n:=\underbrace{\lambda\oplus\cdots\oplus\lambda}_{\times n}.$$

Before we continue let us introduce a very useful result for computing the Möbius function of lattices known as the Crosscut Theorem. We denote the join of $A\subseteq X$ as $\vee A:=\min\{x\in X\,|\,x\ge a\text{ for all } a\in A\}$:

\[prop:crosscut\](Crosscut Theorem, see [@Sta97 Corollary 3.9.4]) Let $L$ be a finite lattice with top element $\hat{1}$ and bottom element $\hat{0}$. Let $X$ be a subset of $L$ such that $\hat{0}\not\in X$ and for all $s\in L$, $s\not=\hat{0}$, there is some $t\in X$ such that $s\ge t$. Then $$\mu(\hat{0},\hat{1})=\sum_{\substack{A\subseteq X\\\vee A=\hat{1}}}(-1)^{|A|}.$$

The Crosscut theorem is traditionally used to compute the Möbius function of a lattice, but we can use it in reverse 