---
tag: 4PR2JBW87G
kind: statement
role: theorem
name: The Reciprocity Theorem
sources:
  - wikipedia: en:1093331@r1330576210
    locator: Stanley's reciprocity theorem, section The Reciprocity Theorem
    note: adopted
    mention: wikipedia://en/1093331#REFS
    sections: [Statement]
inventory: [K460FH4TND]
---

## Statement

Stanley's reciprocity theorem states that for a $d$-dimensional rational cone $C$, we have the following identity of rational functions:

$$F_C(1/x_1,\dots,1/x_d)=(-1)^d F_{{\rm int}(C)}(x_1,\dots,x_d).$$

Stanley's reciprocity theorem generalizes Ehrhart-Macdonald reciprocity for Ehrhart polynomials of rational convex polytopes. Both of these results are examples of combinatorial reciprocity theorems, a term that was, in fact, also coined by Stanley.