The number of M-sequences and f-vectors

We give a recursive formula for the number of M-sequences (a.k.a. f-vectors for multicomplexes or O-sequences) in terms of the number of variables and a maximum degree; In particular, it is shown that the number of M-sequence s for at most 2 variables are powers of two and for at most 3 variables are Bell numbers. We give an asymptotic estimate of the number of M-sequences when the number of variables is fixed. This leads to a new lower bound for the number of polytopes with few vertices. We also prove a similar recursiv e formula for the number of f-vectors for simplicial complexes. Keeping the maximum degree fixed we get the number of M-sequences and the number of f- vectors for simplicial complexes as polynomials in the number of variables and it is shown that these numbers are asymptotically equal.