MathDB

Let

S = \left\{ 1,2,\dots,n \right\}

, where

n \ge 1

. Each of the

2^n

subsets of

S

is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set

T \subseteq S

, we then write

f(T)

for the number of subsets of

T

that are blue.
Determine the number of colorings that satisfy the following condition: for any subsets

T_1

and

T_2

S

f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2).

Problem Statement