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?



Succesful applications of algebra in combinatorics
Stanley's proof of the Upper Bound Conjecture relied on a connection with free resolutions of graded algebras. This has led to the very active area of Stanley--Reisner theory, where combinatorial properties of simplicial complexes are related to algebraic properties of certain graded algebras. 


For references, there's a wikipedia page on Stanley--Reisner theory if you're interested:


<http://en.wikipedia.org/wiki/Stanley%E2%80%93Reisner_ring>


Also, Bruns and Herzog's book "Cohen--Macaulay Rings" has nice a chapter on Stanley--Reisner rings. I'm sure there are other good references as well.


