Hilbert polynomials of configuration spaces over graphs of circumference at most 1

The $k$-configuration space $B_k\Gamma$ of a topological space $\Gamma$ is the space of sets of $k$ distinct points in $\Gamma$. In this paper, we consider the case where $\Gamma$ is a graph of circumference at most $1$. We show that for all $k\ge0$, the $i$-th Betti number of $B_k\Gamma$ is given by a polynomial $P_\Gamma^i(k)$ in $k$, called the Hilbert polynomial of $\Gamma$. We find an expression for the Hilbert polynomial $P_\Gamma^i(k)$ in terms of those coming from the canonical $1$-bridge decomposition of $\Gamma$. We also give a combinatorial description of the coefficients of $P_\Gamma^i(k)$.