, Volume 7, Issue 1, pp 23–34 | Cite as

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

  • Anders Björner
  • Richard Stanley
  • Peter Frankl


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.

AMS subject classification (1980)

05 A 15 55 U 05 


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Copyright information

© Akadémiai Kiadó 1987

Authors and Affiliations

  • Anders Björner
    • 1
  • Richard Stanley
    • 1
  • Peter Frankl
    • 2
  1. 1.Massachusetts Inst. of TechnologyCambridgeUSA
  2. 2.University of Paris, VIIParisFrance

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