Selected Works of Richard P. Stanley

The book under review is a Selecta of works by the prominent combinatorist, Richard Stanley, known in particular for his proof of the Upper Bound Conjecture for spheres: it “gives an explicit upper bound \(f_i(n,d)\) for the number of \(i\)-dimensional faces of a triangulation of \((d-1)\)-dimensional sphere with \(n\) vertices.” In fact, the first article in the present selection is Stanley’s own account of what he did, titled, “How the Upper Bound Conjecture Was Proved,” originally appearing in the Annals of Combinatorics in 2014. The account is positively charming, taking the reader from Stanley’s high school days in Savannah, Georgia, to the 1974 ICM in Vancouver, where he learnt that Reisner’s recent discovery of “Cohen-Macaulay face rings” was exactly the missing piece of the puzzle he needed to complete his proof; says Stanley: “This was the greatest ‘math high’ of my career….”

Of course this is just the tip of the iceberg as far as the book under review is concerned, and we encounter in the 800 or so pages that follow material on (to pick 6, a perfect number): “Supersolvable lattices,” “Combinatorial reciprocity theorems,” “linear Diophantine equations and local cohomology,” “Unimodality and Lie superalgebras,” “A symmetric function generalization of the chromatic polynomial of a graph,” and “A conjectured combinatorial interpretation of the normalized irreducible character values of the symmetric group.” More broadly, the Foreword, by Patricia Hersh, Thomas Lam, Pavlo Pylyavskyy, and Victor Reiner, provides the following description: “This volume contains some of Richard Stanley’s most influential papers. One finds in them a recurring theme: innocent counting problems can reveal deep structure, connecting them to algebra, geometry and topology … We hope that the reader is as surprised, delighted, and inspired by Stanley’s revelations as we have been.” Indeed.

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Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.