In the comments above you ask how to compute these connection numbers. Paul Johnson mentions that you can do this using $S_d$ character theory. I want to also point out his two papers Tropical Hurwitz Numbers and Chamber Structure of Double Hurwitz numbers, written together with Renzo Cavalieri and Hannah Markwig, are very nice and have a lot of explicit formulas.

They are answering a question that is a bit less general than you are asking. That question is the following: Let $\sigma$ be the partition $(2,1,1,\ldots,1)$ of $n$. Let $\lambda$ and $\nu$ be other partitions of $n$. What is the coefficient of $K_{\nu}$ in $K_{\lambda} K_{\sigma}^N$? This isn't quite the same as asking about $K_{\lambda} K_{\mu}$, but you would probably learn a lot from these papers anyways.

In the comments above you ask how to compute these connection numbers. Paul Johnson mentions that you can do this using $S_d$ character theory. I want to also point out his two papers Tropical Hurwitz Numbers and Chamber Structure of Double Hurwitz numbers, written together with Renzo Cavalieri and Hannah Markwig, are very nice and have a lot of explicit formulas.

They are answering a question that is a bit less general than you are asking. That question is the following: Let $\sigma$ be the partition $(2,1,1,\ldots,1)$ of $n$. Let $\lambda$ and $\nu$ be other partitions of $n$. What is the coefficient of $K_{\nu}$ in $K_{\lambda} K_{\sigma}^N$? This isn't quite the same as asking about $K_{\lambda} K_{\mu}$, but you would probably learn a lot from these papers anyways.