chambers of the arrangement $${\mathcal{A}}_{n,m}=\left\{H_{i,j} \mid \alpha_{H_{i,j}}=\lambda_i-j\right\}_{{1 \leqslant i \leqslant n} \atop {0 \leqslant j < m_i-1} }.$$ Namely, $x=v_1\dotsm v_n$ corresponds to the chamber $C$ that contains the open set $$\left\{(\lambda_1,\dotsc ,\lambda_n) \in {\mathbb{R}}^n \mid v_j-1 < \lambda_j < v_j,\ j=1, \dotsc ,n\right\}.$$

#### Salvetti’s complex

There is a correspondence [@MarengoSette] between the oriented graph ${\mathcal{Y}}_{\succ}$ and a subcomplex of the $1$-skeleton of Salvetti’s complex ${\mathcal{S}}({\mathcal{A}}_{n,m})$ as follows. Namely, there is a one-to-one correspondence between the $0$-skeleton ${\mathcal{S}}_0({\mathcal{A}}_{n,m})$ and the set of chambers in ${\mathcal{A}}_{n,m}$, i.e. the set of social outcomes $X$ by means of the correspondence above. The generators of the $1$-skeleton can be described as $${\mathcal{S}}_1({\mathcal{A}}_{n,m})=\{(x,y) \in X \times X \setminus \Delta \mid x\ \text{and}\ y\ \text{are adjacent}\}$$ where two chambers $C$ and $D$ are said to be [*adjacent*]{} if they are separated by only one hyperplane.

Given a subset of consecutive elements $$\{(x_1,x_2),(x_2,x_3),\dotsc, (x_{k-2},x_{k-1}),(x_{k-1},x_k)\}$$ in ${\mathcal{S}}_1({\mathcal{A}}_{n,m})$ their [*formal sum*]{} is $$(x_1,x_k)=\sum_{j=1}^{k-1} (x_j,x_{j+1}).$$ It follows that given a social rule $\succ$ any arc $(x,y) \in {\mathcal{Y}}_{1,\succ}$ can be written as a formal sum of a minimal number of consecutive elements in ${\mathcal{S}}_1({\mathcal{A}}_{n,m})$. The number of elements is exactly the number of hyperplanes that separate the two social 