On The Toric Ideals of the Coloured Graphs of Reduced Words

We study a family $\mathcal{B}$ of pseudo-multipartite graphs indexed by staircase partitions. They are realised from the reduced words of certain class of permutations. We investigate the vertex proper colouring of these graphs and give the general chromatic polynomial. For each member $B_{\lambda}$, we construct an affine toric ideal $\mathcal{I}_{B_{\lambda}}$ associated to vertex proper colouring using partition identity. It turns out that the projective version $\mathcal{V}(\mathcal{I}_{B_{\lambda}})$ is realised from the cartoon diagram associated with the vertex proper colouring.