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We study the gaps $\Delta p_n$ between consecutive rank sizes in$r$-differential posets by introducing a projection operator whose matrixentries can be expressed in terms of the number of certain paths in the Hassediagram. We strengthen Miller's result that $\Delta p_n \geq 1$, which resolveda longstanding conjecture of Stanley, by showing that $\Delta p_n \geq 2r$. Wealso obtain stronger bounds in the case that the poset has many substructurescalled threads.

Path counting and rank gaps in differential posets