Combinatorica

, Volume 7, Issue 1, pp 23–34

# The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem

• Anders Björner
• Richard Stanley
• Peter Frankl
Article

## Abstract

A Cohen-Macaulay complex is said to be balanced of typea=(a1,a2, ...,as) if its vertices can be colored usings colors so that every maximal face gets exactlyai vertices of thei:th color. Forb=(b1,b2, ...,bs), 0≦ba, letfb denote the number of faces havingbi vertices of thei:th color. Our main result gives a characterization of thef-vectorsf=(fb)0≦ba or equivalently theh-vectors, which can arise in this way from balanced Cohen-Macaulay complexes. As part of the proof we establish a generalization of Macaulay’s compression theorem to colored multicomplexes. Finally, a combinatorial shifting technique for multicomplexes is used to give a new simple proof of the original Macaulay theorem and another closely related result.

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### References

1. [1]
A. Björner, Shellable and Cohen-Macaulay partially ordered sets,Trans. Amer. Math. Soc. 260 (1980), 159–183.
2. [2]
A. Björner, Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings,Advan. in Math. 52 (1984), 173–212.
3. [3]
A. Björner, A. Garsia andR. Stanley, An introduction to Cohen-Macaulay partially ordered sets, in:Ordered Sets (ed., I. Rival),Reidel, Dordrecht 1982, 583–615.Google Scholar
4. [4]
P. Frankl, A new short proof for the Kruskal-Katona theorem,Discrete Math. 48 (1984), 327–329.
5. [5]
C. Greene andD. Kleitman, Proof techniques in the theory of finite sets, in:Studies in Combinatorics (ed., G.-C. Rota), Math. Ass. of America, Washington, D. C. 1978, 22–79.Google Scholar
6. [6]
B. Lindström andH.-O. Zetterström, A combinatorial problem in thek-adic number system,Proc. Amer. Math. Soc. 18 (1967), 166–170.
7. [7]
F. S. Macaulay, Some properties of enumeration in the theory of modular systems,Proc. London Math. Soc. 26 (1927), 531–555.
8. [8]
R. P. Stanley, Cohen-Macaulay complexes, in:Higher Combinatorics (ed., M. Aigner), Reidel, Dordrecht/Boston, 1977, 51–62.Google Scholar
9. [9]
R. P. Stanley, Hilbert functions of graded algebras,Advan. in Math. 28 (1978), 57–83.
10. [10]
R. P. Stanley, Balanced Cohen-Macaulay complexes,Trans. Amer. Math. Soc. 249 (1979), 139–157.
11. [11]
R. P. Stanley,Combinatorics and Commutative Algebra, Birkhäuser, Boston, 1983.
12. [12]
G. F. Clements andB. Lindström, A generalization of a combinatorial theorem of Macaulay,J. Comb. Theory,7 (1969), 230–238.
13. [13]
P. Frankl, Z. Füredi andG. Kalai, Shadows of colored complexes,manuscript, 1986.Google Scholar
14. [14]
P. Frankl, The shifting technique in extremal set theory,Proc. British Comb. Coll., London, 1987,in press.Google Scholar