# Combinatorial reciprocity theorems

@article{Stanley1974CombinatorialRT,
title={Combinatorial reciprocity theorems},
author={Richard P. Stanley},
year={1974},
volume={14},
pages={194-253}
}
• R. Stanley
• Published 1 October 1974
• Mathematics
A combinatorial reciprocity theorem is a result which establishes a kind of duality between two related enumeration problems. This rather vague concept will become clearer as more and more examples of such theorems are given. We shall be content in this paper with explaining the meaning of various reciprocity theorems via mere statements of results, together with clarifying examples. A rigorous treatment with detailed proofs appears in [11].
171 Citations
Ehrhart-Macdonald reciprocity extended
• Mathematics
• 2005
For a convex polytope P with rational vertices, we count the number of integer points in integral dilates of P and its interior. The Ehrhart-Macdonald reciprocity law gives an intimate relation
Euler's relation, möbius functions and matroid identities
• Mathematics
• 1982
We give a short combinatorial proof of the Euler relation for convex polytopes in the context of oriented matroids. Using counting arguments we derive from the Euler relation several identities
An Alexander-type duality for valuations
• Mathematics
• 2014
We prove an Alexander-type duality for valuations for certain subcomplexes in the boundary of polyhedra. These strengthen and simplify results of Stanley (1974) and Miller-Reiner (2005). We give a
Meet-distributive lattices and the anti-exchange closure
This paper defines the anti-exchange closure, a generalization of the order ideals of a partially ordered set. Various theorems are proved about this closure. The main theorem presented is that a
A Reciprocity Relation for t-Designs
• Mathematics
Eur. J. Comb.
• 1987
Reciprocal Domains and Cohen-Macaulay d-Complexes in Rd
• Mathematics
Electron. J. Comb.
• 2005
A reciprocity theorem of Stanley about enumeration of integer points in polyhedral cones when one exchanges strict and weak inequalities is extended to include Cohen{Macaulayness and canonical modules.
The Chromatic Quasisymmetric Class Function of a Digraph
• J. White
• Mathematics
Annals of Combinatorics
• 2021
We introduce a quasisymmetric class function associated to a group acting on a double poset or on a directed graph. The latter is a generalization of the chromatic quasisymmetric function of a

## References

SHOWING 1-10 OF 36 REFERENCES
A decomposition for combinatorial geometries
A construction based on work by Tutte and Grothendieck is applied to a decomposition on combinatorial pregeometries in order to study an important class of invariants. The properties of this Tutte
Introduction to Combinatorial Analysis
This book introduces combinatorial analysis to the beginning student. The author begins with the theory of permutation and combinations and their applications to generating functions. In subsequent
Acyclic orientations of graphs
A note on linear homogeneous diophantine equations
in which the fa are arbitrary integers. In this paper this conjecture is proved by induction. Since this solution is written down directly from (1) and is fully displayed these results are more
Natural Partial Orders
• Mathematics
• 1968
Let n be an ordinal. A partial ordering P of the ordinals T = T(n) = {w: w < n} is called natural if x P y implies x ⩽ y. A natural partial ordering, hereafter abbreviated NPO, of T(n) is thus a
Convex Polytopes
Graphs, Graphs and Realizations Before proceeding to the graph version of Euler’s formula, some notation will be introduced. A (finite) abstract graph G consists of two sets, the set of vertices V =
The Tutte polynomial
$q$-Matroids are defined on complemented modular support lattices. Minors of length 2 are of four types as in a "classical" matroid. Tutte polynomials $\tau(x,y)$ of matroids are calculated either by
The volume of a lattice polyhedron
• I. MacDonald
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 1963
Let L be the lattice of all points with integer coordinates in the real affine plane R2 (with respect to some fixed coordinate system). Let X be a finite rectilinear simplicial complex in R2 whose
Sur les partitions non croisees d'un cycle
• G. Kreweras
• Computer Science, Mathematics
Discret. Math.
• 1972