Segre embeddings, Hilbert series and Newcomb's problem
Abstract
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 in I, in some special situations the monomial ideal in 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]⊂ 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.
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