In this brief note a straightforward combinatorial proof for an identity directly connecting rooted forests and unordered set partitions is provided. Furthermore, references that put this type of identity in the context of and are given.

Let and be two unit vectors in a normed plane . We say that is to if the line through in the direction supports the unit disc. A (Fankhänel in Beitr Algebra Geom 52(2):335-342, 2011) is an angular measure on the unit circle for which whenever is a shorter arc of the unit circle connecting two Birkhoff orthogonal points. We present a characterization of the normed planes that admit a B-measure.

An orthogonality space is a set together with a symmetric and irreflexive binary relation. Any linear space equipped with a reflexive and anisotropic inner product provides an example: the set of one-dimensional subspaces together with the usual orthogonality relation is an orthogonality space. We present simple conditions to characterise the orthogonality spaces that arise in this way from finite-dimensional Hermitian spaces. Moreover, we investigate the consequences of the hypothesis that an orthogonality space allows gradual transitions between any pair of its elements. More precisely, given elements and , we require a homomorphism from a divisible subgroup of the circle group to the automorphism group of the orthogonality space to exist such that one of the automorphisms maps to , and any of the automorphisms leaves the elements orthogonal to and fixed. We show that our hypothesis leads us to positive definite quadratic spaces. By adding a certain simplicity condition, we furthermore find that the field of scalars is Archimedean and hence a subfield of the reals.

We answer some questions on the asymptotics of ballot walks raised in [S. B. Ekhad and D. Zeilberger, April 2021] and prove that these models are not D-finite. This short note demonstrates how the powerful tools developed in the last decades on lattice paths in convex cones help us to answer some challenging problems that were out of reach for a long time. On the way we generalize tandem walks to the family of large tandem walks whose steps are of arbitrary length and map them bijectively to a generalization of ballot walks in three dimensions.

We find all polyhedral graphs such that their complements are still polyhedral. These turn out to be all self-complementary.

Consider the family of polynomial differential systems of degree 3, or simply cubic systems in the plane . An equilibrium point of a planar differential system is a if there is a neighborhood of such that is filled with periodic orbits. When is filled with periodic orbits, then the center is a . For this family of cubic systems Lloyd and Pearson characterized in Lloyd and Pearson (Comput Math Appl 60:2797-2805, 2010) when the origin of coordinates is a center. We classify which of these centers are global centers.