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

Abstract

A Cohen-Macaulay complex is said to be balanced of typea=(a 1,a 2, ...,a s ) if its vertices can be colored usings colors so that every maximal face gets exactlya i vertices of thei:th color. Forb=(b 1,b 2, ...,b s ), 0≦ba, letf b denote the number of faces havingb i vertices of thei:th color. Our main result gives a characterization of thef-vectorsf=(f b )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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    A. Björner, Shellable and Cohen-Macaulay partially ordered sets,Trans. Amer. Math. Soc. 260 (1980), 159–183.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    A. Björner, Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings,Advan. in Math. 52 (1984), 173–212.

    MATH  Article  Google Scholar 

  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.

    MATH  Article  MathSciNet  Google Scholar 

  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.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    F. S. Macaulay, Some properties of enumeration in the theory of modular systems,Proc. London Math. Soc. 26 (1927), 531–555.

    Article  Google Scholar 

  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.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    R. P. Stanley, Balanced Cohen-Macaulay complexes,Trans. Amer. Math. Soc. 249 (1979), 139–157.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    R. P. Stanley,Combinatorics and Commutative Algebra, Birkhäuser, Boston, 1983.

    Google Scholar 

  12. [12]

    G. F. Clements andB. Lindström, A generalization of a combinatorial theorem of Macaulay,J. Comb. Theory,7 (1969), 230–238.

    MATH  Google Scholar 

  13. [13]

    P. Frankl, Z. Füredi andG. Kalai, Shadows of colored complexes,manuscript, 1986.

  14. [14]

    P. Frankl, The shifting technique in extremal set theory,Proc. British Comb. Coll., London, 1987,in press.

Download references

Author information

Affiliations

Authors

Additional information

First and third authors partially supported by the National Science Foundation.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Björner, A., Stanley, R. & Frankl, P. The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem. Combinatorica 7, 23–34 (1987). https://doi.org/10.1007/BF02579197

Download citation

AMS subject classification (1980)

  • 05 A 15
  • 55 U 05