I am currently interested to compute relative Gromov Witten invariants(GW) over $\mathbb{P}^1$. In the paper https://arxiv.org/pdf/math/0204305.pdf eq 3.1 gives the count of relative disconnected GW invariants let $\mu,\nu$ are partition of $d$ then gives the GW invariants $\mu$ is the ramification profile over say $0$ and $\nu$ over $\infty$. $$\langle \mu,\prod_{i}^{n}\tau_{k_i}(w),\nu\rangle^{\bullet} $$ I could use the formula to count the disconnected GW over $\mathbb{P}^1$. Is it true that if partition $\mu$ if it's one part that is $\mu=[d]$ then the connected GW invariants and disconnected are the same? We can group these invariants as genus $g$ where $g$ is a positive integer. We say a GW invariant is of genus $g$ if $$\sum_i k_i = 2g-2+\ell{\mu}+\ell{\nu}$$ where $\ell{}$ denotes the length of the partition. My calculation doesn't agree for the case calculating genus 1 GW invariants for the case say
Also I got different answer for $$<1|\tau_{2}|1>^{\bullet} = (61009/66355200)$$ and $$<1|\tau_{2}|1>^{\circ} = 1/24$$ and
My main aim, for now, is to compute connected relative GW invariants for small genus and few partitions. Is there is any reference where it's calculated or table? Also, Theorem 3 in the paper gives a possible calculation but I am not getting what exactly the formula means some step of the calculation would help.
