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number of maps and limit

Source: Problem 3, Polish NO 1994

October 7, 2005
limitLaTeXfunctionalgebra unsolvedalgebra

Problem Statement

kk is a fixed positive integer. Let ana_n be the number of maps ff from the subsets of {1,2,...,n}\{1, 2, ... , n\} to {1,2,...,k}\{1, 2, ... , k\} such that for all subsets A,BA, B of {1,2,...,n}\{1, 2, ... , n\} we have f(A∩B)=min⁡(f(A),f(B))f(A \cap B) = \min (f(A), f(B)). Find lim⁡n→∞ann\lim_{n \to \infty} \sqrt[n]{a_n}.
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