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

A common theme of enumerative combinatorics are counting functions given by polynomials that are evaluated at positive integers. Many such counting functions come with a combinatorial reciprocity theorem: a combinatorial function, which is a priori defined on the positive integers, (1) can be algebraically extended beyond the positive integers (e.g., because it is a polynomial), and (2) has (possibly quite different) meaning when evaluated at negative integers. These combinatorial reciprocity theorems are the center pieces--indeed, the gems--of this book. Alongside them, we'll introduce numerous useful and, we hope, attractive concepts from enumerative geometric combinatorics.

A preliminary version of our manuscript is here. We would appreciate hearing about corrections and suggestions!

"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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