Hi people! This is yet another problem from the entrance test to CMI (Chennai Mathematical Institute) (http://www.cmi.ac.in/).

Consider sets \( \displaystyle A = \{1,2,...,k\} \) and \( \displaystyle B = \{1,2,...,n\} \). Denote \( P_k \) as the power set of \( A \). How many functions \( f \) can be defined from \( P_k \) to \( B \) such that \( f( M \cup N ) = \text{max} ( f(M), f(N) ) \)?

Example: For \( k = 2 \), this function is valid:

\( f( \phi ) = 2 \)

\( f(\{1\}) = 3 \)

\( f(\{2\}) = 5 \)

\( f(\{1\} \cup \{2\} ) = f( \{1, 2 \} ) = \text{max} ( f(\{1\}), f(\{2\}) ) = 5 \)

While the following function is invalid:

\( f( \phi ) = 2 \)

\( f(\{1\}) = 3 \)

\( f(\{2\}) = 5 \)

\( f(\{1\} \cup \{2\} ) = f( \{1, 2 \} ) = 3 \)

Your answer must be a formula involving \( n, k \) only. For \( n = 4, k = 3 \), the number of such functions is \( 100 \).

I had fun solving it!

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TopNewestIs it\( \displaystyle \sum_{i=1}^n i^k \)?

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That it is! Great!

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