  Monomial ideals and toric rings are closely related. By consider a Grobner
basis we can always associated to any ideal $I$ in a polynomial ring a monomial
ideal ${\rm in}_\prec I$, in some special situations the monomial ideal ${\rm
in}_\prec I$ is square free. On the other hand given any monomial ideal $I$ of
a polynomial ring $S$, we can define the toric $K[I]\subset S$. In this paper
we will study toric rings defined by Segre embeddings, we will prove that their
$h-$ vectors coincides with the so called Simon Newcomb number's in
probabilities and combinatorics. We solve the original question of Simon
Newcomb by given a formula for the Simon Newcomb's numbers involving only
positive integer numbers.
