Counting Subsets solved by 1615

July 19, 2012, midnight by Rosalind Team

Topics: Combinatorics, Set Theory

Characters and SNPs

A character is any feature (genetic, physical, etc.) that divides a collection of organisms into two separate groups. One commonly used genetic character is the possession of a single-nucleotide polymorphism, or SNP.

In a process called genotyping, the SNP markers taken from a large number of human donors have been used very successfully to catalogue the migration and differentiation of human populations over the last 200,000 years. For $199, you can participate in National Geographic's Genographic Project, and discover your own genetic heritage.

Whether we use genetic or physical characters, we may think of a collection of $n$ characters as a collection of "ON"/"OFF" switches. An organism is said to possess a character in the "ON" position (although often the assignment of "ON"/"OFF" is arbitrary). Given a collection of taxa, we may represent a character by the collection of taxa possessing the character.

Problem

A set is the mathematical term for a loose collection of objects, called elements. Examples of sets include $\{\textrm{the moon, the sun, Wilford Brimley}\}$ and $\mathbb{R}$, the set containing all real numbers. We even have the empty set, represented by $\emptyset$ or $\{\}$, which contains no elements at all. Two sets are equal when they contain the same elements. In other words, in contrast to permutations, the ordering of the elements of a set is unimportant (e.g., $\{\textrm{the moon, the sun, Wilford Brimley}\}$ is equivalent to $\{\textrm{Wilford Brimley, the moon, the sun}\}$). Sets are not allowed to contain duplicate elements, so that $\{\textrm{Wilford Brimley, the sun, the sun}\}$ is not a set. We have already used sets of 2 elements to represent edges from a graph.

A set $A$ is a subset of $B$ if every element of $A$ is also an element of $B$, and we write $A \subseteq B$. For example, $\{\textrm{the sun, the moon}\} \subseteq \{\textrm{the sun, the moon, Wilford Brimley}\}$, and $\emptyset$ is a subset of every set (including itself!).

As illustrated in the biological introduction, we can use subsets to represent the collection of taxa possessing a character. However, the number of applications is endless; for example, an event in probability can now be defined as a subset of the set containing all possible outcomes.

Our first question is to count the total number of possible subsets of a given set.

Given: A positive integer $n$ ($n \leq 1000$).

Return: The total number of subsets of $\{1, 2, \ldots, n\}$ modulo 1,000,000.

Sample Dataset

3

Sample Output

8

Hint

What does counting subsets have to do with characters and "ON"/"OFF" switches?

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