Counting chemical compositions using Ehrhart quasi-polynomials

Abstract

To count the number of chemical compositions of a particular mass, we consider an alphabet \({\mathop{\bf a}}\) with a mass function which assigns a mass to each letter in \({\mathop{\bf a}}\) . We then compute the mass of a word (an ordered sequence of letters) by adding the masses of the constituent letters. Our main interest is to count the number of words that have a particular mass, where we ignore the order of the letters within the word. We show first that counting the number of words of a given mass has a geometric interpretation, whose solutions are called Ehrhart quasi-polynomials, a class of functions defined on integers. These special functions are “periodic” in the sense that they use the same polynomial every λ steps. In addition to discovering the connection between counting compositions and Ehrhart quasi-polynomials, we also find number theoretic results that greatly reduce the number of candidates for the period, λ. Finally, we illustrate the usefulness of these results and the use of a software library named barvinok (by Verdoolaege et al.) by applying them to eight different classes of chemical compositions, including organic molecules, peptides, DNA, and RNA.