[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that the inverse of the adjacency matrix of [math]P(k)[/math], given by

[[math]] A_k(\pi,\sigma)=\begin{cases} 1&{\rm if}\ \pi\leq\sigma\\ 0&{\rm if}\ \pi\not\leq\sigma \end{cases} [[/math]]

is the Möbius matrix of [math]P[/math], given by [math]M_k(\pi,\sigma)=\mu(\pi,\sigma)[/math].