On the existence of Hamiltonian paths in the cover graph of \(M\)(\(n\)) (Q1868855)

The poset \(M(n)\) has as its elements the \(n\)-tuples of integers \({\mathbf a}=(a_1,a_2,\ldots,a_n)\) satisfying \(0=a_1=\ldots =a_j<a_{j+1}<\ldots <a_n\leq n\) for some \(j\geq 0\). The order relation is defined by \({\mathbf a}\leq{\mathbf b}\) if and only if \(a_i\leq b_i\) for \(1\leq i\leq n\). The authors show that the cover graph \(M(n)\) has a Hamiltonian path if and only if \({n+1 \choose 2}\) is odd and \(n\neq 5\).