Distinct Sequence

You are given a **binary** string $S$ of length $2\cdot N$. A partition of string $S$ into strings $P$ and $Q$ is said to be *valid* if: - Both $P$ and $Q$ are [subsequences](https://en.wikipedia.org/wiki/Subsequence) of $S$ of length $N$; - Each character of string $S$ belongs to **exactly one** subsequence out of $P$ and $Q$; - Strings $P$ and $Q$ are **NOT** equal. Let $A$ denote the set o