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 $F_N^{(0)}(alpha)= prod_{k=1}^{N-1}(1-{Nover k}alpha)$. 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 $S_3$.
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