  The computation of the $N$-cycle brownian paths contribution $F_N(\alpha)$ to
the $N$-anyon partition function is adressed. A detailed numerical analysis
based on random walk on a lattice indicates that $F_N^{(0)}(\alpha)=
\prod_{k=1}^{N-1}(1-{N\over k}\alpha)$. In the paramount $3$-anyon case, one
can show that $F_3(\alpha)$ is built by linear states belonging to the bosonic,
fermionic, and mixed representations of $S_3$.
