Hi. This may be a very general question.

Are there any examples of problems in combinatorics which were open, but which found a solution when stated in algebraic terms?

If yes, could somebody mention some of these? I'm new to this and don't know many examples yet.

I know about the "Magic Squares", which refers to counting the number of $n\times n$ $\mathbb{N}$-matrices having line sum equal to $r$. This was treated by Anand, Dumir and Gupta, by stating it as the number of ways of distributing $n$ different things, each one replicated $r$ times, among $n$ different persons, in equal numbers. It was solved by R. Stanley (see "Commutative algebra arising from the Anand-Dumir-Gupta conjectures" by Winfried Bruns).

Are there some instances where algebra has been used to enumerate, say, certain sets of graphs?