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A fun problem - find the formula of number of functions from a power set to another set.

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:

  1. \( f( \phi ) = 2 \)

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

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

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

While the following function is invalid:

  1. \( f( \phi ) = 2 \)

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

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

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

    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!

Note by Parth Thakkar
3 years ago

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Is it\( \displaystyle \sum_{i=1}^n i^k \)? Siddhartha Srivastava · 3 years ago

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@Siddhartha Srivastava That it is! Great! Parth Thakkar · 3 years ago

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Other problems I found interesting:

  1. Polynomials? That sounds familiar

  2. And you thought limits were always easy

Parth Thakkar · 3 years ago

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