Embeddability of the combinohedron (Q1613559)

Let \(m \in \mathbb N\) and \(R=\{r_1, \dots, r_m\} \in \mathbb N^m,\) where \(r_i \in \mathbb N\) and \(r_1\geq \cdots \geq r_m.\) Let \(e_1, \dots, e_m\) be \(m\) different symbols and let \(n = \sum_{i=1}^mr_i.\) The combinohedron is the loopless graph whose vertex set corresponds to all \(n\)-tuples in which the symbol \(e_i\) appears exactly \(r_i\) times and where an edge joins vertices \((u_1, \dots, u_n)\) and \((v_1, \dots, v_n)\) if and only if there exists an index \(k\) \((1\leq k < n)\) such that \(u_k = v_{k+1}\), \(u_{k+1} = v_{k},\) and \(u_i = v_i\) for each \(i\) with \(1\leq i \leq n\) and \(i \not= k\), \(k+1.\) The permutohedron is a particular case of the combinohedron by setting \(r_i = 1\) for all \(i\); see \textit{P. Gaiha} and \textit{S. K. Gupta} [SIAM J. Appl. Math. 32, 323-327 (1977; Zbl 0354.05024)]. Extending some results on embeddability of the permutohedron, the authors here discuss some explicit embeddings of the combinohedron into the cubic and root lattices which are closely related with the order-dimension of the multinomial lattice; see \textit{M. K. Bennett} and \textit{G. Birkhoff} [Algebra Univers. 32, 115-144 (1994; Zbl 0810.06006)].