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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?

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    $\begingroup$ Do you include linear algebra? LOTS of graph theory have been done with its use. For example, Tutte's original proof of his matching theorem uses determinants of matrices over polynomial rings (not just $\mathbb Q$ or $\mathbb F_2$). $\endgroup$ May 19 '11 at 11:22
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    $\begingroup$ There are entire books with the title "Algebraic Graph Theory" :-) $\endgroup$ May 19 '11 at 11:57
  • $\begingroup$ there are also books named Algebraic Combinatorics, Algebraic Combinatorics on Words... $\endgroup$ May 19 '11 at 13:44
  • $\begingroup$ I can't tell if you want to ask a very general question or a somewhat more specific one (judging from your last sentence). Could you be slightly more specific about what kind of examples you're looking for and what you're hoping to get out of having a list of such examples? $\endgroup$ May 19 '11 at 13:59
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    $\begingroup$ Nice reading : www.tau.ac.il/~nogaa/PDFS/null2.pdf google.fr/… $\endgroup$ May 19 '11 at 14:39
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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.

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    $\begingroup$ In the same vein is Stanley's proof of the g-theorem, which exploited a connection between rational polytopes (which are combinatorial objects) and toric varieties. Strictly speaking this is a connection between combinatorics and algebraic geometry rather than with pure algebra, but I think it falls within the spirit of the question. $\endgroup$ May 19 '11 at 14:55
  • $\begingroup$ @Timothy: Thanks for the comment. I was unaware of this result. $\endgroup$ May 19 '11 at 16:23

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