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A poset or partially ordered set is a set endowed with a partial order, meaning a binary relation $\leq$ which is reflexive ($x \leq x$ for all $x$), antisymmetric ($x\leq y$ and $y\leq x$ implies $x=y$), and transitive ($x\leq y$ and $y\leq z$ implies $x \leq z$).

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Number of elements in poset with same rank such that lower bound has a certain rank

I was wondering if anything is known about this problem. Fix $0\leq m\leq k$. We are given a graded poset and we fix an element $x$ of rank $k$. Is it possible to estimate the number of elements $y$ o …
Marcel K. Goh's user avatar