Calculate $$\sum_{\sigma \in \mathfrak{S}_n} \varepsilon(\sigma), \quad \sum_{\sigma \in \mathfrak{S}_n} \varepsilon(\sigma) \nu(\sigma) \quad \text{and} \quad \sum_{\sigma \in \mathfrak{S}_n} \frac{\varepsilon(\sigma)}{\nu(\sigma)+1}.$$