0P7H Lemma 6.2.2. Let σ∈Hom𝒮n(I,J)\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J). We have ℓ(σ)=∑0≤i1<i2<ni1,i2∈I~|⌊σ(i2)−σ(i1)n⌋|.\ell(\sigma)=\sum_{\begin{subarray}{c}0\leq i_{1}<i_{2}<n\\ i_{1},i_{2}\in\tilde{I}\end{subarray}}\bigl|{\lfloor\frac{\sigma(i_{2})-\sigma(i_{1})}{n}\rfloor}\bigr|.