We study Defant and Kravitz’s generalization of Schützenberger’s promotion operator to arbitrary labelings of finite posets in two directions. Defant and Kravitz showed that applying the promotion operator \({n-1}\) times to a labeling of a poset on n elements always gives a natural labeling of the poset and called a labeling tangled if it requires the full \({n-1}\) promotions to reach a natural labeling. They also conjectured that there are at most \({(n-1)!}\) tangled labelings for any poset on n elements. We propose a strengthening of their conjecture by partitioning tangled labelings according to the element labeled \({n-1}\) and prove that this stronger conjecture holds for inflated rooted forest posets and a new class of posets called shoelace posets. We also introduce sorting generating functions and cumulative generating functions for the number of labelings that require k applications of the promotion operator to give a natural labeling. We prove that the coefficients of the cumulative generating function of the ordinal sum of antichains are log-concave and obtain a refinement of the weak order on the symmetric group.
