# 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
5 years, 8 months ago

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## Comments

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Is it$\displaystyle \sum_{i=1}^n i^k$?

- 5 years, 8 months ago

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That it is! Great!

- 5 years, 8 months ago

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Other problems I found interesting:

- 5 years, 8 months ago

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