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Abstract
We study the gaps Δpn between consecutive rank sizes in r-differential posets by introducing a projection operator whose matrix entries can be expressed in terms of the number of certain paths in the Hasse diagram. We strengthen Miller’s result that Δpn ≥ 1, which resolved a longstanding conjecture of Stanley, by showing that Δpn ≥ 2r. We also obtain stronger bounds in the case that the poset has many substructures called threads.
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