Counting chemical compositions using Ehrhart quasi-polynomials

To count the number of chemical compositions of a particular mass, we consider an alphabet with a mass function which assigns a mass to each letter in . 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 lambda 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, lambda. 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.