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

The $k$-configuration space $B_k\mathrm{\Gamma }$ of a topological space $\mathrm{\Gamma }$ is the space of sets of $k$ distinct points in $\mathrm{\Gamma }$. In this paper, we consider the case where $\mathrm{\Gamma }$ is a graph of circumference at most $1$. We show that for all $k\ge 0$, the $i$-th Betti number of $B_k\mathrm{\Gamma }$ is given by a polynomial $P_{\mathrm{\Gamma }}^i(k)$ in $k$, called the Hilbert polynomial of $\mathrm{\Gamma }$. We find an expression for the Hilbert polynomial $P_{\mathrm{\Gamma }}^i(k)$ in terms of those coming from the canonical $1$-bridge decomposition of $\mathrm{\Gamma }$. We also give a combinatorial description of the coefficients of $P_{\mathrm{\Gamma }}^i(k)$.