# Map from a power set to n-set (CMI Entrance 2014 Solution)

(1) Let A = {1, … , k} and B = {1, … , n}. Find the number of maps from A to B .
(2) Define $$\mathbf{ P_k }$$ be the set of subsets of A. Let f be a map from $$\mathbf{P_k to B }$$ such that if $$\mathbf{ U , V \in P_k }$$ then $$\mathbf{ f(U \cup V) }$$= $$\mathbf{\text{max} { f(U) , f(V) } }$$ . Find the number of such functions. (For example if k = 3 and n =4 then answer is 100)

Discussion:

(1) For each member x of set A we have n choices for f(x) in B. Hence the number of functions is $$mathbf{ n^k }$$

(2) Claim (i): $$\mathbf{ f(\phi) }$$ is minimum for any such function f from $$\mathbf{P_k to B }$$ . This is because $$\mathbf{ f(A_1) = f(\phi \cup A_1 ) = \text{max} { f(A_1), f(\phi) } }$$ hence $$\mathbf{f(A_1)}$$ must be larger than $$\mathbf{ f(\phi) }$$ for any member $$\mathbf{A_1}$$ of $$\mathbf{ P_k }$$

Claim (ii) If we fix the values of the singleton sets then the entire function is fixed. That is if we fix the values of f({1}) , f({2}) , … , f({k}). Since for any member $$\mathbf{A_1}$$ of $$\mathbf{P_k , {A_1} }$$ is union of several singleton sets. Hence it’s value is the maximum of the functional values of those singleton sets. For example let $$\mathbf{ A_1 = (1, 2) }$$ then $$\mathbf{f(A_1) = f({1}\cup{2}) = \text {max} {f({1}) , f({2}) } }$$

Claim (iii) f({1}) , … , f({k}) are individually independent of each other.

Now we fix $$\mathbf{ f(\phi) = i}$$. Since it is the smallest, the singleton sets map to i to n.
Hence if $$\mathbf{ f(\phi) = 1}$$ each of the singleton sets have n choices from 1 to n; hence there are $$\mathbf{ n^k }$$ functions. Similarly $$\mathbf{ f(\phi) = 2}$$ each of the singleton sets have n-1 choices from 2 to n; hence there are $$\mathbf{ (n-1)^k }$$ functions.

Thus the total number of functions = $$\mathbf{ \sum_{i=1}^n i^k }$$