The g-theorem matrices are totally nonnegative

The g-theorem proved by Billera, Lee, and Stanley states that a sequence is the g-vector of a simplicial polytope if and only if it is an M-sequence. For any d-dimensional simplicial polytope the face vector is g M d where M d is a certain matrix whose entries are sums of binomial coefficients. Bjِrner found refined lower and upper bound theorems by showing that the ( 2 × 2 )-minors of M d are nonnegative. He conjectured that all minors of M d are nonnegative and that is the result of this note.