0P7F Lemma 6.2.1. We have ℓ(σ′∘σ)≤ℓ(σ′)+ℓ(σ)\ell(\sigma^{\prime}\circ\sigma)\leq\ell(\sigma^{\prime})+\ell(\sigma) for all σ∈Hom𝒮n(I,J)\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(I,J) and σ′∈Hom𝒮n(J,K)\sigma^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}_{n}}(J,K).
0P7G Proof. We have L(σ′∘σ)={(i1,i2)∈I~2|i1<i2,σ(i1)>σ(i2),σ′∘σ(i1)>σ′∘σ(i2)}⊔{(i1,i2)∈I~2|i1<i2,σ(i1)<σ(i2),σ′∘σ(i1)>σ′∘σ(i2)}={(i1,i2)∈L(σ)|σ′∘σ(i1)>σ′∘σ(i2)}⊔(σ−1×σ−1)({(j1,j2)∈L(σ′)|σ−1(j1)<σ−1(j2)}).L(\sigma^{\prime}\circ\sigma)=\{(i_{1},i_{2})\in\tilde{I}^{2}\ |\ i_{1}<i_{2},\ \sigma(i_{1})>\sigma(i_{2}),\ \sigma^{\prime}\circ\sigma(i_{1})>\sigma^{\prime}\circ\sigma(i_{2})\}\sqcup\\ \{(i_{1},i_{2})\in\tilde{I}^{2}\ |\ i_{1}<i_{2},\ \sigma(i_{1})<\sigma(i_{2}),\ \sigma^{\prime}\circ\sigma(i_{1})>\sigma^{\prime}\circ\sigma(i_{2})\}\\ =\{(i_{1},i_{2})\in L(\sigma)\ |\ \sigma^{\prime}\circ\sigma(i_{1})>\sigma^{\prime}\circ\sigma(i_{2})\}\sqcup(\sigma^{-1}\times\sigma^{-1})\bigl(\{(j_{1},j_{2})\in L(\sigma^{\prime})\ |\ \sigma^{-1}(j_{1})<\sigma^{-1}(j_{2})\}\bigr). It follows that ℓ(σ′)+ℓ(σ)−ℓ(σ′∘σ)=2|{(i1,i2)∈I~2|i1<i2,σ(i1)>σ(i2),σ′∘σ(i1)<σ′∘σ(i2)}/n𝐙|≥0.\ell(\sigma^{\prime})+\ell(\sigma)-\ell(\sigma^{\prime}\circ\sigma)=2|\{(i_{1},i_{2})\in\tilde{I}^{2}\ |\ i_{1}<i_{2},\ \sigma(i_{1})>\sigma(i_{2}),\ \sigma^{\prime}\circ\sigma(i_{1})<\sigma^{\prime}\circ\sigma(i_{2})\}/n{\mathbf{Z}}|\geq 0. ∎