Stanley's reciprocity theorem
Encyclopedia
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 (defined below) and the generating function of the cone's interior.
A rational cone is the set of all d-tuple
s
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 Fint(x1, ..., xd) 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 function
s. Stanley's reciprocity theorem states that
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".
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Stanley's reciprocity theorem, named after MIT mathematician Richard P. Stanley
Richard P. Stanley
Richard Peter Stanley is the Norman Levinson Professor of Applied Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. He received his Ph.D. at Harvard University in 1971 under the supervision of Gian-Carlo Rota...
, states that a certain functional equation
Functional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
is satisfied by the generating function
Generating function
In mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
of any rational cone (defined below) and the generating function of the cone's interior.
A rational cone is the set of all d-tuple
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
s
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 Fint(x1, ..., xd) 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 function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s. Stanley's reciprocity theorem states that
Matthias Beck, Mike Develin
Mike Develin
Michael Lee Develin is an American mathematician known for his work in combinatorics and discrete geometry.-Early life:...
, and Sinai Robins have shown how to prove this by using the calculus of residues
Methods of contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.Contour integration is closely related to the calculus of residues, a methodology of complex analysis....
. Develin has said that this amounts to proving the result "without doing any work".