In combinatorial mathematics, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley, states that a certain functional equation is satisfied by the generating function of any "rational cone" and the generating function of the cone's interior. Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ... Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Mapúa Institute of Technology (MIT, MapúaTech or simply Mapúa) is a private, non-sectarian, Filipino tertiary institute located in Intramuros, Manila. ... Richard P. Stanley (born 1944) is Norman Levinson Professor of Applied Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. ... In mathematics or its applications, a functional equation is an equation in terms of independent variables, and also unknown functions, which are to be solved for. ... In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...

A "rational cone" is the set of all d-tuples In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects. ...

(a₁, ..., a_d)

of nonnegative integers satisfying a system of inequalities

where M is a matrix of integers. A d-tuple satisfying the corresponding strict inequalities, i.e., with ">" rather than "≥", is in the interior of the cone.

The generating function of such a cone is

The generating function F_int(x₁, ..., x_d) of the interior of the cone is defined in the same way, but one sums over d-tuples in the interior rather than in the whole cone.

It can be shown that these are rational functions. Stanley's reciprocity theorem states that In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

Matthias Beck, Mike Develin, and Sinai Robins have shown how to prove this by using the calculus of residues. Develin has said that this amounts to proving the result "without doing any work". In complex analysis, the evaluation of integrals of real-valued functions along intervals on the real line, is not readily found with certain integrands and methods involving only real variables. ...

References

• R.P. Stanley, "Combinatorial reciprocity theorems", Advances in Mathematics, volume 14 (1974), pages 194 - 253.

External links

• On Stanley's reciprocity theorem for rational cones, by Beck, Develin, and Robbins -- arXiv abstract