vector ā” requires some very delicate geometrical arguments. The part proved
by Stanley ā” that conversely to every simplicial polytope there corresponds
an M -sequence in the stated way ā” uses tools from algebraic geometry in
an essential way. Here is a brief statement for readers with suļ¬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 diļ¬erent mathe-
matical objects, diļ¬erent mathematical problems and diļ¬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 ā” two seemingly totally unrelated classes of objects.
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 Eulerā™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
ā¢ Handbook of Combinatorics (eds. R. Graham, M. GrĀØtschel and L.
LovĀ“sz), North-Holland, Amsterdam, 1995.
A good reference for number partitions is look for
ā¢ G.E. Andrews, Theory of partitions, ...
The basic theory of enumeration is developed in
ā¢ 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 MĀØbius function, together with algebraic ram-
iļ¬cations, is discussed there. A briefer account of this material is given in
ā¢ I. Gessel and R.P. Stanley, Algebraic Enumeration, a chapter in the
āHandbook of Combinatoricsā, pp. xxxā“yyy
A wealth of information about the topic of tilings can be found in
ā¢ GrĀØnbaum and Shepard
For enumerative aspects of tilings see
ā¢ Elkies-Kuperberg-Larsen-Propp paper in J. Alg. Comb.
The following book is a nice companion to the study of enumeration
ā¢ N. J. A. Sloane and S. Plouļ¬e, The Encyclopedia of Integer Sequences,
Academic Press, 1995. There is also an interactive version on the web
For connections between combinatorics and topology, including more details
about the evasiveness and Kneser conjectures, see either of
ā¢ A. BjĀØrner, Combinatorics and Topology, Notices of the American Math-
ematical Society 32 (1985), 339ā“345.
ā¢ A. BjĀØrner, Topological Methods, a chapter in the āHandbook of Com-
binatoricsā, pp. 1819ā“1872.
The disproof of the Borsuk conjecture is reported in
ā¢ B. Cipra, Disproving the obvious in higher dimensions, Whatā™s Hap-
pening in the Mathematical Sciences 1 (1993), 21ā“25,
ā¢ A. Skopenkov, Borsukā™s problem, Quantum 7 (1996), 17ā“21,
while more about the k-equal problem and its solution can be found in
ā¢ A. BjĀØrner, Subspace Arrangements, in āFirst European Congress of
Mathematics, Paris 1992ā (eds. A. Joseph et al), Progress in Mathe-
matics Series, Volume 119, BirkhĀØuser, Boston, 1994, pp. 321ā“370.
Finally, for convex polytopes and the g-theorem we refer to
ā¢ G. M. Ziegler, Lectures on Polytopes, GTM Series, Springer-Verlag,