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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 $$100$$.

Note by Parth Thakkar
3 years ago

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