Degrees of Freedom Versus Dimension for Containment Orders
نویسندگان
چکیده
Given a family of sets S, where the sets in S admit k ‘degrees of freedom’, we prove that not all (k + 1)-dimensional posets are containment posets of sets in S. Our results depend on the following enumerative result of independent interest: Let P (n, k) denote the number of partially ordered sets on n labeled elements of dimension k. We show that logP (n, k) ∼ nk log n where k is fixed and n is large.
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