I'm looking for combinatoric problems that can be solved by counting eulerian circuits in directed graphs, i.e. by defining some graph and then counting all cycles passing through each edge exactly once.
One example I found is (a slight modification of) this question, where we have $n$ people belonging to $k$ disjunct groups of equal size, and who shall be seated around a round-table.
We now want to count the arrangements so that
no pair of two people belonging to the same group have the same left-side-neighbor.
However, this example is quite particular, so I'd love to see some more general examples that illustrate when counting eulerian circuits can be helpful.
BEST Theorem, a polynomial method to count eulerian circuits in digraphs.
$0000111100101101$ is a string of $2^4$ bits with the property that any consecutive $4$ bits (including wrapping around to the front) represents a different $4$-bit binary string. How many such strings are there with that property? Generalize to $n\neq 4$.