Number of connected sets of size $k$ containing a given vertex on $\mathbb{Z}^d$

Question

Consider the graph $\mathbb{Z}^d$. Let $C_k$ be the number of connected sets containing the origin and $k-1$ other vertices. How does $C_k$ grow with $k$?

Answer 1

You're asking (up to some scaling factor) for the number of $d$-dimensional polycubes of size $n$. Barequet, Barequet, and Rote have written about this. It's known that it grows like $\lambda_d^n$ for some constant $\lambda_d$. In the $d = 2$ case it's known that $\lambda_2 \in [4.0025, 4.5276]$ (this is called "Klarner's constant"); numerically it appears to be around 4.062570 (see e. g. this presentation of Knuth). Barequet, Barequet, and Rote show that $\lambda_d = 2ed - o(d)$ as $d \to \infty$. But nobody seems to have written down any numerical estimates for $\lambda_3$, at least nobody I could find quickly.