The number of \(M\)-sequences and \(f\)-vectors
The author gives a recursive formula for the number of \(M\)-sequences in terms of the number of variables and maximum degree. Given a multicomplex (i.e. a multiset system satisfying \(A \subset B \in {\mathcal M} \Rightarrow A \in {\mathcal M}\)) let \(m_i=| \{A \in {\mathcal M}: \deg A=i\}| \). The sequence \(m=(m_1,m_2,\dots)\) is the \(M\)-sequence of \({\mathcal M}\). It is shown that the numbers of \(M\)-sequences for at most 2 variables are powers of two and for at most 3 variables they are Bell numbers. An asymptotic estimate of the number of \(M\)-sequences is also given in the case when the number of variables is fixed. This estimate is used to show lower bounds for the number of polytopes with few vertices. A similar recursive formula is proven for the number of \(f\)-vectors of simplicial complexes.
scientific article; zbMATH DE number 1381104