Zero forcing in graphs is a coloring process where a vertex colored blue can force its unique uncolored neighbor to be colored blue. A zero forcing set is a set of initially blue vertices capable of eventually coloring all vertices of the graph. In this paper, we focus on the numbers z(G;i), which is the number of zero forcing sets of size i of the graph G. These numbers were initially studied by Boyer et al. [5] where they conjectured that for any graph G on n vertices, z(G;i)≤z(Pn;i) for all i≥1 where Pn is the path graph on n vertices. The main aim of this paper is to show that several classes of graphs, including outerplanar graphs and threshold graphs, satisfy this conjecture. We do this by studying various graph operations and examining how they affect the number of zero forcing sets.

The collection of reflecting hyperplanes of a finite Coxeter group is called a reflection arrangement and it appears in many subareas of combinatorics and representation theory. We focus on the problem of counting regions of reflection arrangements and their deformations. Inspired by the recent work of Bernardi, we show that the notion of moves and sketches can be used to provide a uniform and explicit bijection between regions of (the Catalan deformation of) a reflection arrangement and certain non-nesting partitions. We then use the exponential formula to describe a statistic on these partitions such that distribution is given by the coefficients of the characteristic polynomial. Finally, we consider a sub-arrangement of type C arrangement called the threshold arrangement and its Catalan and Shi deformations.

It is known that for the Young diagram of any partition of an integer n, the sum of squares of the hook lengths of its cells is exactly n(2) morethan that of the contents of its cells. That is, for any partition lambda of an integer n, & sum;(u is an element of lambda )h(u)(2 )= n(2 )+ & sum;(u is an element of lambda )c(u)(2).We provide a bijective proof of this fact, thus solving a problem posed by Stanley. Along the way, we obtain a formula for the number of rectangles in the Young diagram of a partition. We also mention a result for sums of other powers of hook lengths and contents.