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

Example: For k=2 k = 2 , this function is valid:

  1. f(ϕ)=2 f( \phi ) = 2

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

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

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

While the following function is invalid:

  1. f(ϕ)=2 f( \phi ) = 2

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

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

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

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

I had fun solving it!

Note by Parth Thakkar
5 years, 4 months ago

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1 vote

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Is iti=1nik \displaystyle \sum_{i=1}^n i^k ?

Siddhartha Srivastava - 5 years, 4 months ago

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That it is! Great!

Parth Thakkar - 5 years, 4 months 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 - 5 years, 4 months ago

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