We consider Hilbert series associated to modules over various categories of trees. Using the technology Sam and Snowden (J Am Math Soc 30(1): 159–203, 2017) we show that these must be algebraic. then apply technical theorems prove facts about certain natural generating functions

Generalizing a result of [24] , [8] about elliptic modular forms, we give closed formula for the sum all Hilbert Hecke eigenforms over totally real number field with strict class 1, multiplied by their period polynomials, as single product Kronecker series.

For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis ...

For certain subsets S and T of A = { · · · , (0, 2), (0, 1), (0, 0), (1, 0), (2, 0), · · · } and factor spaces C S [X, Y ], C +S,T [X, Y, Z,W ] and C − S,T [X, Y, Z,W ], bitableaux bases are constructed that are indexed by pairs of standard tableaux and sequences in the collections ΥψS and ΥψT . These bases give combinatorial interpretations to the appropriate Hilbert series of these spaces as ...