Math Graduate Student Seminar

We study the gaps Δp_n 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 Δp_n≥1, which resolved a longstanding conjecture of Stanley, by showing that Δp_n≥2r. We also obtain stronger bounds in the case that the poset has many substructures called threads. (joint work with Christian Gaetz)