Hi all, 


I am doing a work in collaboration with other mathematicians about phase transition in the Ising model and we need to know if exponential upper bounds exist for the number of lattice animals with boundary of size $n$. 


To be precise, consider the square lattice $\mathbb{Z}^2$ as graph where the edges are pairs of points in the lattice having distance one from each other, where the distance is induced by the norm $\|(z\_1,z\_2)\|=|z\_1|+|z\_2|$. 


We call a lattice animal the set of vertices of any connected subgraph of the square lattice. Given an animal $A$, we denote the boundary of $A$ by $\partial A$, that is, the set of vertices of distance one from $A$. 


Fix a site $z\in \mathbb{Z}^2$ and let be  

$$
f(n)=\sharp \{A\ \text{is lattice animal}; A\ni z\ \text{and}\ |\partial A|=n\}
$$  

Is it known if $f(n)=O(e^{k n})$ ?


I learned from google that this problem is also known in the combinatorics community as enumeration of polyominoes with a given site-perimeter. 


All the papers I found about the upper bounds at some point have to impose some geometric hypotheses on the polyominoes such as convexity, starcase shape or bargraph shape. 


I don't know yet if those hypotheses are being used in order to get sharp upper bounds or if they are the only ones available. 


If the question about exponential upper bound is not yet solved, is there a specialist in this area who could tell me what they think about the upper bound for this problem.



Exponential bounds for the number of lattice animals with a given boundary. 
I think the following (or something close) proves Leandro’s point:


Let $n=5k$ for some odd integer $k$, and let $B\_k$ be the subgraph of $\mathbb{Z}^2$ with vertex set $[1\dots k]\times[1\dots k]$ (and all edges between these vertices from the lattice $\mathbb{Z}^2$). Let K be any subset of $[2\dots k-1]\times[2\dots k-1]$ of size $k$ in which every point has both coordinates even. Let $A\_k^K$ be the subgraph of $B\_k$ obtained by removing the points of $K$ from the vertex set. The graph $A\_k^K$ is an animal, and the boundary of $A\_k^K$ has cardinality $n$; the boundary comprises the $4k$ points exterior to and adjacent to $B\_k$ (they remain adjacent to $A\_k$, because no points on the perimeter of $B\_k$ were removed) together with the $k$ points of $K$, each of which is adjacent to a location with at least one odd coordinate. The animals $A\_k^K$ and $A\_k^{K'}$ are distinct if $K\neq K'$. 


Let $\mathcal{A}\_k$ be set of $A\_k^K$ for all possible sets $K$ described above. Then $f(n) \ge |\mathcal{A}\_k|$. (Let $z=(1,1)$). The number of distinct sets $K$ is approximately $k^2 /4\choose k$, which grows faster than $k! = (n/5)!$ and is therefore not bounded by an exponential. 


