There is a strikingly simple classical formula for the number of lattice paths avoiding the line x = ky when k is a positive integer.We show that the natural generalization of this simple formula continues to hold when the line x = ky is replaced by certain periodic staircase boundaries-but only under special conditions. The simple formula fails in general, and it remains an open question to what extent our results can be further generalized.
We investigate a number of questions concerning representations of a set of numbers as sums of subsets of some other set. In particular, we obtain several results on the possible sizes of the second set when the first set consists of a geometric sequence of integers, partially answering a generalisation of a question of Gerry Myerson.
Academic Press
In graph pegging, we view each vertex of a graph as a hole into which a peg can be placed, with checker-like "pegging moves" allowed. Motivated by well-studied questions in graph pebbling, we introduce two pegging quantities. The pegging number (respectively, the optimal pegging number) of a graph is the minimum number of pegs such that for every (respectively, some) distribution of that many pegs on the graph, any vertex can be reached by a sequence of pegging moves. We prove several basic properties of pegging and analyze the pegging number and optimal pegging number of several classes of graphs, including paths, cycles, products with complete graphs, hypercubes, and graphs of small diameter.
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