Sufficient Conditions for a Symmetric Chain Order

It is shown that a partial order satisfying the “LYM property”, symmetry of Whitney numbers under the inversion j → n - j, and unimodality of Whitney numbers can be partitioned into symmetric chains. It follows that the same conclusion follows if the LYM property assumption is replaced by the condition that every element of rank k is ordered with the same number of elements of ranks $k - 1$ and $k + 1$. It further follows that the lattice of subspaces of finite-dimensional vector space over a finite field is a symmetric chain order.