An explicit formula for the number of permutations with a given number of alternating runs

Let $R(n,k)$ denote the number of permutations of ${1,2,...,n}$ with $k$ alternating runs. In this note we present an explicit formula for the numbers $R(n,k)$.Comment: 6 page

A Dynamic Adoption Model with Bayesian Learning: Application to the U.S. Soybean Market

Agricultural technology adoption is often a sequential process. Farmers may adopt a new technology in part of their land first and then adjust in later years based on what they learn from the earlier partial adoption. This paper presents a dynamic adoption model with Bayesian learning, in which forward-looking farmers learn from their own experience and from their neighbors about the new technology. The model is compared to that of a myopic model, in which farmers only maximize their current benefits. We apply the analysis to a sample of U.S. soybean farmers from year 2000 to 2004 to examine their adoption pattern of a newly developed genetically modified (GM) seed technology. We show that the myopic model predicts lower adoption rates in early years than the dynamic model does, implying that myopic farmers underestimate the value of early adoption. My results suggest that farmers in my sample are more likely to be forward-looking decision makers and they tend to rely more on learning from their own experience than learning from their neighbors.technology adoption, Bayesian learning, structural estimation, Agribusiness, Agricultural and Food Policy, Crop Production/Industries, Industrial Organization, Production Economics, Research and Development/Tech Change/Emerging Technologies, Research Methods/ Statistical Methods, Risk and Uncertainty, Teaching/Communication/Extension/Profession,