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One of the major outstanding conjectures in the study of chromatic symmetric functions (CSF's) states that trees are uniquely determined by their CSF's. Though verified on graphs of order up to twenty-nine, this result has been proved only for certain subclasses of trees. Using the definition of the CSF that emerges via the Frobenius character map applied to C[Sn], we offer new algebraic proofs of several results about the CSF's of trees. Additionally, we prove that a "parent function" of the CSF defined in the group ring of Sn can uniquely determine trees, providing further support for Stanley's conjecture.

We study the prime numbers that lie in Beatty sequences of the form ⌊αn + β⌋ and have prescribed algebraic splitting conditions. We prove that the density of primes in both a fixed Beatty sequence with α of finite type and a Chebotarev class of some Galois extension is precisely the product of the densities α −1 · |C| |G| . Moreover, we show that the primes in the intersection of these sets satisfy a Bombieri-Vinogradov type theorem. This allows us to prove the existence of bounded gaps for such primes. As a final application, we prove a common generalization of the aforementioned bounded gaps result and the Green-Tao theorem.

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