Enumeration of pyramids of one-dimensional pieces of arbitrary fixed integer length

We consider pyramids made of one-dimensional pieces of fixed integer length a and which may have pairwise overlaps of integer length from 1 to a. We prove that the number of pyramids of size m, i.e. consisting of m pieces, equals (am-1,m-1) for each a >= 2. This generalises a well known result for a = 2. A bijective correspondence between so-called right (or left) pyramids and a-ary trees is pointed out, and it is shown that asymptotically the average width of pyramids is proportional to the square root of the size

Environments-by-Design: Developing new tools to bring “field conditions” to the laboratory

Maximizing agricultural yields is an urgent priority for our society as the world population will increase to 9 billion by 2050. Additionally, climate change could have detrimental effects on food supply and biomass production. Addressing these problems means considering the relationship the environment has on a plant’s development. The reality is though that there are few tools to plant scientists that can properly control and monitor the growing environment for plant. We believe that as engineers we can provide plant scientists with new tools which we call “environments by design”. These tools can control the environmental stimuli either abiotic (e.g. water availability, nutrient concentration, temperature, relative humidity) or biotic (e.g. microbes or other plants) delivered to plant in time and space. By controlling the environment, hypotheses of how genetic traits interact with environmental factors to yield the plant phenotypes can be addressed inside the laboratory