I was trying to look into the following problem, which I intend to use for a lemma for a bigger problem.
The question is: For the 2-dimensional integer lattice, what are some good lower and upper bounds for the number of connected components of size $k$ that contain the origin? (We say that two lattice points are connected if they are the two endpoints of an edge)
I tried to tackle the 1-dimensional case, but for that, the answer is much easier since you can actually compute the total number of such connected components.
I also tried to write a program that computes it and it seems like the boundaries should be some exponentials, but I didn't manage to find a proof for this.
Thanks
The number of connected components of size $k$ that contain the origin is exactly the number $P_f(k)$ of fixed $k$-minoes multiplied by $k$. Clearly, $P(k)\le P_f(k)\le 8P(k)$, where $P(k)$ is number of $k$-polyominoes. The best currently known bounds on $P(k)$ are $3.72^k< P(k) <4.65^k$.