Simultaneous orderings and combinatorial interpretations of generalized factorials and binomial coefficients

Abstract

In this paper, we answer several questions raised by Manjul Bhargava concerning p-orderings, p-sequences, and combinatorial interpretations of generalized factorials and binomial coefficients associated with subsets S of $Z$. First, we prove some results of Bhargava concerning p-orderings, p-sequences and the integrality of generalized binomial coefficients directly from definitions, answering two questions of Bhargava. As a result, we also obtain some explicit descriptions of the p-sequences associated with p-orderings which do not exist in Bhargava's proofs. Second, we provide some combinatorial interpretations for the generalized factorials and the generalized binomial coefficients associated with subsets of $Z$ which possess simultaneous p-orderings, answering two other questions raised by Bhargava.