I have a question about the $\psi$ class in the following paper of Graber and Vakil: https://arxiv.org/abs/math/0309227
For $k,d\geq 2$, and a partition $d=d_1+\cdots+d_k$ of $d$ into positive integers, let $$\overline{\mathcal{M}}\equiv \overline{\mathcal{M}}_{0,(d_1,\cdots,d_k;d)}(\mathbb{P}^1,0\cup \infty)_{\sim}$$ denote the relative moduli space of genus zero degree $d$ coverings of $\mathbb{P}^1$ with ramification orders $d_1,\ldots,d_k$ over $0$ and $d$ over $\infty$, up to the $\mathbb{C}^*$-action on $\mathbb{P}^1$ (That's what the subscript $\sim$ mean). The moduli space $\overline{\mathcal{M}}$ is a smooth orbifold of complex dimension $k-2$. A generic map in $\overline{\mathcal{M}}$ is a polynomial $(z-z_1)^{d_1}\cdots(z-z_k)^{d_k}$ defined over $\mathbb{P}^1$ with the relative marked points $(z_1,\ldots,z_k,\infty)$. The complement of this open set consists of maps to a chain of $\mathbb{P}^1$'s. This gives a map from $\overline{\mathcal{M}}$ to $\mathcal{T}_{\sim}$ which is the Artin stack parametrizing the possible non-rigid targets (chains of $\mathbb{P}^1$). The latter is the open substack of $\mathcal{M}_{0,2}$ parametrizing $2$-pointed rational curves where the only nodes separate $0$ from $\infty$.
Let $\psi$ be the $\psi$-class in Graber-Vakil's paper, defined in Section 2.5. It is the pullback to $\overline{\mathcal{M}}$ of $\psi_0$ over $\mathcal{T}_{\sim}$.
Question:
- Is the value of $\int_{\overline{\mathcal{M}}}\psi^{k-2}$ calculated somewhere?
I browsed Graber-Vakil, Faber-Pandharipande, etc. The space $\overline{\mathcal{M}}$ and the integral above appear in Lemma 4.6-4.8 of Graber-Vakil but I have not been able to find the numbers above explicitely.
