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Motivated by the study of integer partitions, we consider partitions ofintegers into fractions of a particular form, namely with constant denominatorsand distinct odd or even numerators. When numerators are odd, the numbers ofpartitions for integers smaller than the denominator form symmetric patterns.If the number of terms is restricted to $h$, then the nonzero terms of thegenerating function are unimodal, with the integer $h$ having the mostpartitions. Such properties can be applied to a particular class of nonlinearDiophantine equations. We also examine partitions with even numerators. Weprove that there are $2^{\omega(t)}-2$ partitions of an integer $t$ intofractions with the first $x$ consecutive even integers for numerators and equaldenominators of $y$, where $0<y<x<t$. We then use this to produce corollariessuch as a Dirichlet series identity and an extension of the prime omegafunction to the complex plane, though this extension is not analyticeverywhere.

Counting Restricted Partitions of Integers into Fractions: Symmetry and Modes of the Generating Function and a Connection to $\omega(t)$