The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem
- Cite this article as:
- Björner, A., Stanley, R. & Frankl, P. Combinatorica (1987) 7: 23. doi:10.1007/BF02579197
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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≦b≦a, letfb denote the number of faces havingbi vertices of thei:th color. Our main result gives a characterization of thef-vectorsf=(fb)0≦b≦a′ 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.