A Generating Polynomial for Convex Subsets of Posets

For any finite poset we define a generating polynomial counting upsets, downsets, and their intersection. We investigate the behaviour of this polynomial with respect to poset operations and show that it distinguishes certain posets constructed with the operations of ordinal sum and disjoint union, including Series-Parallel posets as a special case. Lastly, we comment on connections to the causal set approach to quantum gravity.