User nicolae boicu - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T19:05:35Z http://mathoverflow.net/feeds/user/24805 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22462/what-are-some-examples-of-interesting-uses-of-the-theory-of-combinatorial-species/100997#100997 Answer by Nicolae Boicu for What are some examples of interesting uses of the theory of combinatorial species? Nicolae Boicu 2012-06-30T13:43:36Z 2012-06-30T13:43:36Z <p>Since the Theory of species is also about relabeling, and relabellings form permutation groups, the Theory of Species is very close related to Permutation Groups theory. Is the theory of species a part of group theory ? NO. And if it were, someone should take it out of there and make it a standalone theory. </p> <p>Imagine that someone wants to write a book containing equations like Part = E(E+), or that amazing Joyal' spines on Cayley formula. Then the author must write a lot of permutations stuff, leaving the formulas for the last chapter maybe, and confusing the reader: Is this book about permutations or about something else ?</p> <p>The functorial definition avoid all the permutation trouble that would have been involved, including the definition of species on empty sets or the notion of copies of the empty set. A simple object like an empty box is not likely described by the mathematical empty set. Anyway, if there are still troubles with the empty sets and the void permutations, these are less visible in cats language. </p> <p>Hence I think the main worry of the authors was to not reload the permutation group theory - and this is highly understandable. All the pieces of this mega combinatorial puzzle were already know : Burnside rings, wreath product, the fix of an element, Polya's polynomials on symmetries and the exponential generating functions. I also think that any presentation of Species should emphasize somehow the magic of egf calculus, that is for Combinatorics what the O and 1 calculus is for the true and false Logic. </p> <p>Today, when many e.g.f.'s are listed, anyone could observe some "mysterious" relationships between egf's and it could build at least some mnemonic meanings that bring some orders in huge lists formulas. The Theory of Species tries to give a scientific base to this collection of mnemonic meanings. It is like someone invents the classical synthetic geometry after two thousands years of analytic Cartesian geometry and he try to well found it. </p> <p>Bibliography <a href="http://www.math.sinica.edu.tw/www/file_upload/mayeh/1989Therelationsbetweenpermutation.pdf" rel="nofollow">http://www.math.sinica.edu.tw/www/file_upload/mayeh/1989Therelationsbetweenpermutation.pdf</a> Labelle and Yeh in 1987 on Permutation groups and Species to all Species fans as I am -> I am also watching the talk page on wikipedia <a href="http://en.wikipedia.org/wiki/Talk:Combinatorial_species" rel="nofollow">http://en.wikipedia.org/wiki/Talk:Combinatorial_species</a></p>