Combinatorial Reciprocity Theorems: An Invitation To Enumerative Geometric Combinatorics is a book by Matthias Beck and Raman Sanyal published in 2018 in the Graduate Studies in Mathematics series of the American Mathematical Society.

A combinatorial reciprocity is a curious phenomenon: A polynomial, whose evaluations at positive integers count combinatorial objects of some sort, may give the number of combinatorial objects of a different sort when evaluated at negative integers (and suitably normalized). Such combinatorial reciprocity theorems occur in connections with graphs, partially ordered sets, polyhedra, and more. With combinatorial reciprocity theorems as a leitmotif, this book unfolds central ideas and techniques in enumerative and geometric combinatorics.

Written as an accessible graduate textbook with almost 300 exercises, numerous illustrations, and pointers to the research literature, topics include concise introductions to partially ordered sets, polyhedral geometry, and rational generating functions, followed by highly original chapters on subdivisions, geometric realizations of posets, and hyperplane arrangements.

The final version of our manuscript is here.

"First, it is necessary to study the facts, to multiply the number of observations, and then later to search for formulas that connect them so as thus to discern the particular laws governing a certain class of phenomena. In general, it is not until after these particular laws have been established that one can expect to discover and articulate the more general laws that complete theories by bringing a multitude of apparently very diverse phenomena together under a single governing principle."
Augustin Louis Cauchy

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