Question 190
The Bank of Oslo issues coins made out of two types of metal: aluminium (denoted $A$) and copper (denoted $C$). Morgane has $n$ aluminium coins, and $n$ copper coins, and arranges her $2n$ coins in a row in some arbitrary initial order. Given a fixed positive integer $k \leqslant 2n$, she repeatedly performs the following operation: identify the largest subsequence containing the $k$-th coin from the left which consists of consecutive coins made of the same metal, and move all coins in that subsequence to the left end of the row. For example, if $n=4$ and $k=4$, the process starting from the configuration $A A C C C A C A$ would be
$$
A A C C C A C A \rightarrow C C C A A A C A \rightarrow A A A C C C C A \rightarrow C C C C A A A A \rightarrow \cdots .
$$
Find all pairs $(n, k)$ with $1 \leqslant k \leqslant 2n$ such that for every initial configuration, at some point of the process there will be at most one aluminium coin adjacent to a copper coin.
$$
A A C C C A C A \rightarrow C C C A A A C A \rightarrow A A A C C C C A \rightarrow C C C C A A A A \rightarrow \cdots .
$$
Find all pairs $(n, k)$ with $1 \leqslant k \leqslant 2n$ such that for every initial configuration, at some point of the process there will be at most one aluminium coin adjacent to a copper coin.