# Special Year Research Seminar

## The Factorial Function and Generalizations, Revisited.

In 1996 Manjul Barghava introduced a notion of P-orderings for arbitrary sets S of a Dedekind domain, with respect to a prime ideal P, which defined associated invariants called P-sequences. He combined these invariants to define generalized factorials and binomial coefficients associated to the subset S. These factorials were used in characterizing rings of polynomials that are integer-valued on S. Further generalizations of P-orderings by Bhargava in 2009 (with more parameters) have many applications. This talk defines analogous invariants for all proper ideals B of a Dedekind domain, called B-sequences, and extends the notion of generalized factorials and binomial coefficients to this setting. (This is joint work with Wijit Yangjit (U. Michigan))