An n-arc in a projective plane is a collection of n distinct points in the plane, no three of which lie on a line. Formulas counting the number of n-arcs in any finite projective plane of order q are known for n ≤ 8. In 1995, Iampolskaia, Skorobogatov, and Sorokin counted 9-arcs in the projective plane over a finite field of order q and showed that this count is a quasipolynomial function of q. We present a formula for the number of 9-arcs in any projective plane of order q, even those that are non-Desarguesian, deriving Iampolskaia, Skorobogatov, and Sorokin's formula as a special case. We obtain our formula from a new implementation of an algorithm due to Glynn; we give details of our implementation and discuss its consequences for larger arcs.
Our results in this paper are threefold: First, we establish the modular properties of the graded dimensions of principal subspaces of level one standard modules for [Formula: see text], and of principal subspaces of certain higher level standard modules for [Formula: see text]. Second, we establish the modular properties of families of q-series that appear in identities due to Warnaar and Zudilin, which generalize Macdonald's [Formula: see text] identities and the Rogers–Ramanujan identities. Third, we formulate a number of conjectures regarding the modularity of series of this type related to AN-1 root systems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.