DOIAbstract
We find an explicit formula for the number of graded partially ordered sets of rank h that can be defined on a set containing n elements. Also, we find the number of graded partially ordered sets of length h, and having a greatest and least element that can be defined on a set containing n elements. The first result provides a lower bound for G*(n), the number of posets that can be defined on an n-set; the second result provides an upper bound for the number of lattices satisfying the Jordan-Dedekind chain condition that can be defined on an n-set
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