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

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Work out the full proof of the explicit formula for the Weingarten function for [math]S_N[/math], namely

[[math]] W_{kN}(\pi,\nu)=\sum_{\tau\leq\pi\wedge\nu}\mu(\tau,\pi)\mu(\tau,\nu)\frac{(N-|\tau|)!}{N!} [[/math]]

then of the main estimate for this function, namely

[[math]] W_{kN}(\pi,\nu)=N^{-|\pi\wedge\nu|}( \mu(\pi\wedge\nu,\pi)\mu(\pi\wedge\nu,\nu)+O(N^{-1})) [[/math]]

where [math]\mu[/math] is the Möbius function of [math]P(k)[/math].