vector ” requires some very delicate geometrical arguments. The part proved

by Stanley ” that conversely to every simplicial polytope there corresponds

an M -sequence in the stated way ” uses tools from algebraic geometry in

an essential way. Here is a brief statement for readers with su¬cient back-

ground. There are certain complex projective varieties, called toric varieties,

associated to d-polytopes with rational coordinates, and the fact that the

sequence g corresponding to the f -vector of a polytope is an M -sequence

ultimately derives from a multicomplex that can be constructed in the coho-

mology algebra of such a variety.

The g-vector associated to a simplicial polytope via the g-theorem is

rich in geometric, algebraic and combinatorial meaning, yet it is still poorly

understood and the subject of much current study.

In this paper we have several times commented on the many surprising,

remarkable and mysterious connections that exist between di¬erent mathe-

matical objects, di¬erent mathematical problems and di¬erent mathematical

areas. Take for example the Schensted correspondence described in Section

3, connecting permutations and pairs of standard Young tableaux; or the

connections between combinatorics and representation theory or combina-

torics and topology described in earlier sections. The g-theorem is one more

example of this kind, establishing an unsuspected link between the combina-

torial structure of multicomplexes of monomials and the facial structure of

simplicial polytopes ” two seemingly totally unrelated classes of objects.

88

In closing, let us once more mention that no characterization is known

for f -vectors of general polytopes of dimension greater than 3. The success

in the case of simplicial polytopes depends on some very special structure,

available in that case but lacking or much more complex in general. Thus,

the study of f -vectors, initiated by Euler™s discovery almost 250 years ago,

is likely to remain an important challenge for many years to come.

13 Further reading (incomplete)

We refer here mainly to general accounts that should be at least partially

accessible to the layman and that give lots of further references.

For a broad view of current combinatorics, with a wealth of information

and references, see

• Handbook of Combinatorics (eds. R. Graham, M. Gr¨tschel and L.

o

Lov´sz), North-Holland, Amsterdam, 1995.

a

A good reference for number partitions is look for

popular

articles

• G.E. Andrews, Theory of partitions, ...

The basic theory of enumeration is developed in

• R.P. Stanley, Enumerative Combinatorics, Volume 1 (Wadsworth &

Brooks/Cole, Monterey, CA, 1986) and Volume 2 (Cambridge Univer-

sity Press, Cambridge, UK, to appear in 1997/8?).

The combinatorics of number and set partitions, standard Young tableaux,

generating functions and the M¨bius function, together with algebraic ram-

o

i¬cations, is discussed there. A briefer account of this material is given in

89

• I. Gessel and R.P. Stanley, Algebraic Enumeration, a chapter in the

“Handbook of Combinatorics”, pp. xxx“yyy

A wealth of information about the topic of tilings can be found in

• Gr¨nbaum and Shepard

u

For enumerative aspects of tilings see

• Elkies-Kuperberg-Larsen-Propp paper in J. Alg. Comb.

The following book is a nice companion to the study of enumeration

• N. J. A. Sloane and S. Plou¬e, The Encyclopedia of Integer Sequences,

Academic Press, 1995. There is also an interactive version on the web

at http://www.research.att.com/∼njas/sequences/

For connections between combinatorics and topology, including more details

about the evasiveness and Kneser conjectures, see either of

• A. Bj¨rner, Combinatorics and Topology, Notices of the American Math-

o

ematical Society 32 (1985), 339“345.

• A. Bj¨rner, Topological Methods, a chapter in the “Handbook of Com-

o

binatorics”, pp. 1819“1872.

The disproof of the Borsuk conjecture is reported in

• B. Cipra, Disproving the obvious in higher dimensions, What™s Hap-

pening in the Mathematical Sciences 1 (1993), 21“25,

• A. Skopenkov, Borsuk™s problem, Quantum 7 (1996), 17“21,

90

while more about the k-equal problem and its solution can be found in

• A. Bj¨rner, Subspace Arrangements, in “First European Congress of

o

Mathematics, Paris 1992” (eds. A. Joseph et al), Progress in Mathe-

matics Series, Volume 119, Birkh¨user, Boston, 1994, pp. 321“370.

a

Finally, for convex polytopes and the g-theorem we refer to

• G. M. Ziegler, Lectures on Polytopes, GTM Series, Springer-Verlag,

Berlin, 1995.

91