This week we are celebrating in Cambridge MA , and elsewhere in the world, Richard Stanley’s birthday. For the last forty years, Richard has been one of the very few leading mathematicians in the area of combinatorics, and he found deep, profound, and fruitful links between combinatorics and other areas of mathematics. His works enriched and

influenced combinatorics as well as other areas of mathematics, and, in my opinion,

combinatorics matured greatly as a mathematical discipline thanks to his work.

## Trivia Quiz

**Correct or incorrect?**

(1) Richard drove cross-country at least 8 times

(2) In his youth, at a wild party, Richard Stanley found a proof of FLT consisting of a few mathematical symbols.

(3) Richard jumped at least once from an airplane

(4) Richard is actively interested in the study of **consciousness**

(5) Richard found a mathematical way to divide by zero

## Seven Early Papers by Richard Stanley That You Must Read.

**Richard’s Green book: Combinatorics and Commutative Algebra**

### Combinatorics and Commutative Algebra

(1) R. P. Stanley, The upper bound conjecture and Cohen-Macaulay rings.

Studies in Appl. Math. 54 (1975), no. 2, 135–142.

The two seminal papers (1) and (3) (below) showed remarkable and unexpected applications of commutative algebra to combinatorics. In each of these papers a central

conjecture in combinatorics was solved in a completely unexpected way which was the basis for a later remarkable theory. Paper (1) is the starting point for the interrelation between commutative algebra and combinatorics of simplicial complexes and their

topology. In this work Richard Stanley proved the Motzkin-Klee upper bound conjecture for triangulations of spheres. This conjecture asserts that the maximum number

of *k*-faces for a triangulation of a *(d-1)*-dimensional sphere with *n* vertices is attained by the boundary complex of the cyclic *d*-dimensional polytope with

*n* vertices. Peter McMullen proved this conjecture for simplicial polytopes and Richard Stanley proved it for arbitrary triangulations of spheres. The key point was that a certain ring (the Stanley-Reisner ring) associated with a simplicial polytope has the *Cohen-Macaulay property*.

The connection between combinatorics and commutative algebra is

far reaching, and in subsequent works combinatorial problems led to

developments in commutative algebra and techniques from the two areas were

combined. A more recent important paper by Richard on applications of commutative algebra for the study of face numbers is: R. P. Stanley, Subdivisions and local *h*-vectors. J. Amer. Math. Soc. 5 (1992), no. 4, 805–851.

And here is, a few weeks old important development in this theory: Relative Stanley-Reisner theory and Upper Bound Theorems for Minkowski sums, by Karim A. Adiprasito and Raman Sanyal.

### The Cohen-Macaulay property, magic squares and lattice points in polytopes

(2) R. P. Stanley, Magic labelings of graphs, symmetric magic squares,

systems of parameters, and Cohen-Macaulay rings. Duke Math. J. 43 (1976),

no. 3, 511–531.

This paper starts with a theorem about enumeration of certain magic squares. Solving a long-standing open problem, Stanley proved that the generating function for the

number of *k* by *k* integer matrices (*k*– fixed) with nonnegative entries and row sums and column sums equal to *n*is rational. This is the starting

point of a deep algebraic theory of integral points in polyhedra.

### Enters the Hard-Lefshetz Theorem: McMullen’s* g*-conjecture

(3) R. P. Stanley, The number of faces of a simplicial convex polytope. Adv. in Math. 35 (1980), no. 3, 236–238.

The *g*-conjecture proposes a complete characterization of face numbers of d-dimensional polytopes. One linear equality that holds among face numbers is, of course, the Euler-Poincaré relation. This relation implies additional [d/2] equalities called the Dehn-Sommerville relations. Peter McMullen proposed an additional system of linear and nonlinear inequalities as a complete characterization of face numbers of polytopes. The sufficiency part of this conjecture was proved by Billera and Lee. Richard Stanley’s brilliant proof for McMullen’s inequalities that established the *g*-conjecture was based on the Hard Lefschetz Theorem from algebraic topology. Starting from a simplicial polytope *P* (with rational vertices) we associate to it a toric variety *T(P)*. It turned out that the cohomology ring of this variety is closely related to the Stanley-Reisner ring mentioned above. The Hard Lefschetz Theorem implies an algebraic property of the Stanley-Reisner ring from which McMullen inequalities can be deduced by direct combinatorial reasoning. Richard found a number of other combinatorial applications of the Hard Lefschetz theorem (including the solution of the Erdos-Moser conjecture).

Here is the abstract of Lou Billera’s lecture

**LOUIS BILLERA (CORNELL)
**

**Even more intriguing, if rather less plausible…**

The title is how Peter McMullen described his own conjectured characterization of the f-vectors of simplicial polytopes in his 1971 lecture notes on the upper bound conjecture written with Geoffrey Shephard. Yet by the end of that decade, the so-called g-conjecture would become the g-theorem, and algebraic combinatorics (as practiced at MIT) would have attracted the attention of mainstream mathematics, almost entirely due to the startling proof given by Richard Stanley.

I will briefly describe some of the events leading to this proof and some of its still developing consequences.

### Enumeration

Enumeration is Richard’s true mathematical love.

Richard’s monumental books EC1 and EC2 (The picture is of EC1 and a young fan)

(4) A baker’s dozen of conjectures concerning plane partitions, in *Combinatoire Énumérative* (G. Labelle and P. Leroux, eds.), Lecture Notes in Math., no. 1234, Springer-Verlag, Berlin/Heidelberg/New York, 1986, pp. 285-293.

13 beautiful conjectures on counting plane partitions with various forms of symmetry.

(5) Generating functions, in *Studies in Combinatorics* (G.-C. Rota, ed.), Mathematical Association of America, 1978, pp 100-141.

For me this was the best introduction to generating functions, clear and inspiring. The entire MAA 1978 Rota’s blue little volume on combinatorics is great. Buy it!

### Posets everywhere

(5) Supersolvable lattices, *Algebra Universalis* **2** (1972), 197-217.

This paper provides a profound link between group theory and the study of

partially ordered sets. It can be seen as a starting point of Stanley’s own work on the Cohen-Macaulay property and it had much influence on later works on combinatorial properties of lattices of subgroups by Quillen and many others, and also on the study of POSETS (=partially ordered sets) arising from arrangements of hyperplanes. The algebraic notion of supersolvable groups is translated to an important combinatorial notion for partially ordered sets. (There is a more detailed paper which I could not find online: R. P. Stanley, Supersolvable semimodular lattices. Mobius algebras (Proc. Conf., Univ. Waterloo, Waterloo, Ont., 1971), pp. 80–142. Univ. Waterloo, Waterloo, Ont., 1971.)

### Combinatorics and representation theory

(6) On the number of reduced decompositions of elements of Coxeter groups, *European J. Combinatorics* **5** (1984), 359-372.

This paper gives an important result proved using representation theory. It is one of

many results by Stanley on connections between enumerative combinatorics,

representation theory, and invariant theory. Again, this paper represents an exciting area of research about the connection of enumerative combinatorics and representation theory that I am less familiar with. A very inspiring survey paper is: Invariants of finite groups and their applications to combinatorics, *Bull. Amer. Math. Soc.* (new series) **1** (1979), 475-511.

Here are abstracts of two lectures from the meeting on some recent developments in combinatorial representation theory and symmetric functions.

**GRETA PANOVA (UCLA)
**

**The Kronecker coefficients: an unexpected journey**

Kronecker coefficients live at the intersection of representation theory, algebraic combinatorics and, most recently, complexity theory. They count the multiplicities of irreducible representations in the tensor product of two other irreducible representations of the symmetric group. While their journey started 75 years ago, they still haven’t found their explicit positive combinatorial formula, and present a major open problem in algebraic combinatorics. Recently, they were given a new role in the field of Geometric Complexity Theory, initiated by Mulmuley and Sohoni, where certain conjectures on the complexity of computing and deciding positivity of Kronecker coefficients are part of a program to prove the “”P vs NP”” problem.

We will take the Kronecker coefficients to asymptotics land and bound them. As an unexpected consequence of this trip, we find bounds for the difference of consecutive coefficients in the q-binomial coefficients (as polynomial in q), generalizing Sylvester’s unimodality theorem and connecting with results of Richard Stanley.

Joint work with Igor Pak.

**THOMAS LAM (U MICHIGAN)
**

**Truncations of Stanley symmetric functions and amplituhedron cells**

Stanley symmetric functions were invented (by Stanley) with applications to the enumeration of reduced words in the symmetric group in mind. Recently, the “amplituhedron” was introduced in the study of scattering amplitudes in N=4 super Yang Mills. I will talk about a formula for the cohomology class of a (tree) amplituhedron variety as the truncation of an affine Stanley symmetric function.

### Two combinatorial applications of the Aleksandrov-Fenchel inequalities;

(7) Two combinatorial applications of the Aleksandrov-Fenchel inequalities, *J. Combinatorial Theory (A)* **31** (1981), 56-65.

In this amazing paper Stanley used inequalities of classical convexity

to settle an important conjecture on probability of events in partially

ordered sets. A special case of the conjecture was settled earlier by Ron

Graham using the FKG inequality. The profound relation between classical

convexity inequalities, combinatorial structures, polytopes, and

probability theory was further studied by many authors including Stanley

himself and there is much more to be done.

I see that I ran out of my seven designated slots. Certainly you should read Richard’s combinatorial constructions of polytopes, like Two poset polytopes, *Discrete Comput. Geom.* **1** (1986), 9-23, and his papers on arrangements. Let me mention a more recent paper of Stanley in this general area: A polytope related to empirical distributions, plane trees, parking functions, and the associahedron (with J. Pitman), *Discrete Comput. Geom.*, **27** (2002), 603-634.

# More

(Mostly from RS’s homepage.)

### Chess and Mathematics

(12 page PDF file) An excerpt (version of 1 November 1999) from a book Richard is writing with Noam Elkies.

### An unusual method for proving the Riemann hypothesis.

### Richard Stanley’s Mathoverflow question on MAGIC

### Magic trick based on deep mathematics

## Richard the Catalan

(From RS’s homepage)

Excerpt (27 page PDF file) from EC2 on problems related to Catalan numbers (including 66 combinatorial interpretations of these numbers).

Solutions to Catalan number problems from the previous link (23 page PDF file).

**Catalan addendum** (Postscript or PDF) (version of 25 May 2013; 96 pages). An addendum of new problems (and solutions) related to Catalan numbers. Current number of combinatorial interpretations of *C _{n}*:

**207**.

The material on Catalan numbers is being collected into a monograph, to be published by Cambridge University Press in late 2014 or early 2015.