Succesful applications of algebra in combinatorics - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T03:01:36Z http://mathoverflow.net/feeds/question/65415 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65415/succesful-applications-of-algebra-in-combinatorics Succesful applications of algebra in combinatorics Camilo Sarmiento 2011-05-19T10:35:06Z 2011-05-19T14:36:53Z <p>Hi. This may be a very general question. </p> <p>Are there any examples of problems in combinatorics which were open, but which found a solution when stated in algebraic terms?</p> <p>If yes, could somebody mention some of these? I'm new to this and don't know many examples yet.</p> <p>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). </p> <p>Are there some instances where algebra has been used to enumerate, say, certain sets of graphs?</p> http://mathoverflow.net/questions/65415/succesful-applications-of-algebra-in-combinatorics/65442#65442 Answer by Daniel Erman for Succesful applications of algebra in combinatorics Daniel Erman 2011-05-19T14:36:53Z 2011-05-19T14:36:53Z <p>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. </p> <p>For references, there's a wikipedia page on Stanley--Reisner theory if you're interested:</p> <p><a href="http://en.wikipedia.org/wiki/Stanley%E2%80%93Reisner_ring" rel="nofollow">http://en.wikipedia.org/wiki/Stanley%E2%80%93Reisner_ring</a></p> <p>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.</p>