Richard P. Stanley

Hilbert functions of graded algebras

Let R be a Noetherian commutative ring with identity, graded by the nonnegative integers N. Thus the additive group of R has a direct-sum decomposition R = R, + R, + ..., where RiRi C R,+j and 1 E R, . I f in addition R, is a field K, so that R is a k-algebra, we will say that R is a G-akebra. The assumption that R is Noetherian implies that a G-algebra is finitely generated (as an algebra over k) and that each R, is a finite-dimensional vector space over k. The Hilbe-rt function of R is defined by

Log-Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometrya

A sequence a,, a,, . . . , a, of real numbers is said to be unimodal if for some 0 s j _c n we have a, 5 a , 5 . . 5 ai 2 a,,, 2 . . 2 a,, and is said to be logarithmically concave (or log-concave for short) if a: 2 a, ,a ,+ , for all 1 5 i 5 n 1. Clearly a log-concave sequence of positive terms is unimodal. Let us say that the sequence a,, a,, . . . , a, has no internal zeros if there do not exist integers 0 5 i < j < k 5 n satisfying a, f 0, a, = 0, ak # 0. Then in fact a nonnegative log-concave sequence with no internal zeros is unimodal. The sequence a,, a , , . . . , a, is called symmetric if a, = a,-, for 0 5 i 5 n. Thus a symmetric unimodal sequence a,, a, , . . . , a,, has its maximum at the middle term ( n even) or middle two terms (n odd). We also say that a polynomial a, + a,4 +. . + an

Some combinatorial properties of Jack symmetric functions

has a certain property (such as unimodal, log-concave, or symmetric) if its sequence a,, a,, . . . , a, of coefficients has that property. Our object here is to survey the surprisingly rich variety of methods for showing that a sequence is log-concave or unimodal. For each method we will give examples of its applicability to combinatorially defined sequences that arise naturally from problems in algebra, combinatorics, and geometry. We make no attempt, however, to give a comprehensive account of all work done in this area.

The number of faces of a simplicial convex polytope

If 1, + L, + = n, then write 2+-n or 1% = n. If p is another partition, then write p c J. if pi 6 %, for all i (i.e., if the diagram of 1. contains the diagram of p), If IpL/ = Ii.1 then write p 2 1. (reverse lexicographic order) if either /J = i or the first nonvanishing difference E.,--pi is positive. For instance (writing 2,1, . ..I.. for (Ai, I.,, . . . . /it)), 5 5 41 : 32 5 311 : 221 : 2111 : 11111. Finally write p<i if Ipl=lAl and p1+pL2+ ... +pLi< A1 + A2 + . . . li for all i. Macdonald [M,, p. 63 call the partial ordering < the “natural ordering,” but we will call it the dominance ordering Let x=(x,, x2, . ..) be an infinite set of indeterminates. As in [MI], we

Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property

Let P be a simplicial convex d-polytope with fi = fi(P) faces of dimension i. The vector f(P) = (f. , fi ,..., fdel) is called the f-vector of P. In 1971 McMullen [6; 7, p. 1791 conjectured that a certain condition on a vector f = (f. , fi ,..., fd...J of integers was necessary and sufficient for f to be the f-vector of some simplicial convex d-polytope. Billera and Lee [l] proved the sufficiency of McMullen’s condition. In this paper we prove necessity. Thus McMullen’s conjecture is completely verified. First we describe McMullen’s condition. Given a simplicial convex dpolytope P with f(P) = (f. , fi ,..., f&, define

Invariants of finite groups and their applications to combinatorics

Techniques from algebraic geometry, in particular the hard Lefschetz theorem, are used to show that certain finite partially ordered sets

Differentiably Finite Power Series

Q^X

On the Number of Reduced Decompositions of Elements of Coxeter Groups

derived from a class of algebraic varieties X have the k-Sperner property for all k. This in effect means that there is a simple description of the cardinality of the largest subset of

Some Combinatorial Properties of Schubert Polynomials

Q^X

Decompositions of Rational Convex Polytopes

containing no