论文标题
合并合并的组合模型
A combinatorial model for lane merging
论文作者
论文摘要
一条两条车道的道路接近了一大灯。左车道在交叉路口驶入右侧。车辆一次接近十字路口，有些驾驶员总是选择正确的车道，而另一些驾驶员总是选择较短的车道，优先选择正确的车道来打破领带。车辆的到达顺序可以表示为二进制字符串，零代表驱动程序总是选择正确的车道，而驱动器代表驾驶员选择较短的车道。从每个到达序列中，我们构建一个合并路径，这是由每个汽车选择的车道确定的晶格路径。我们为到达序列中的$ k $ ZEROS的合并路径的数量（n，m）$提供了封闭的公式，以及所有到达序列的右车道的预期长度与$ k $ zeros的所有到达序列的预期长度。证明涉及对安德烈反思原则的改编。还出现了其他有趣的连接，包括：投票号，预期的最大头部或尾巴出现在$ n $ coin flips中，这是最大的多米诺骨牌蛇，可以使用最高$ [n：n：n：n：n：n] $制成的最大的多米诺骨蛇，以及完整的图形$ k_n $的最长轨迹。
A two lane road approaches a stoplight. The left lane merges into the right just past the intersection. Vehicles approach the intersection one at a time, with some drivers always choosing the right lane, while others always choose the shorter lane, giving preference to the right lane to break ties. An arrival sequence of vehicles can be represented as a binary string, where the zeros represent drivers always choosing the right lane, and the ones represent drivers choosing the shorter lane. From each arrival sequence we construct a merging path, which is a lattice path determined by the lane chosen by each car. We give closed formulas for the number of merging paths reaching the point $(n,m)$ with exactly $k$ zeros in the arrival sequence, and the expected length of the right lane for all arrival sequences with exactly $k$ zeros. Proofs involve an adaptation of Andre's Reflection Principle. Other interesting connections also emerge, including to: Ballot numbers, the expected maximum number of heads or tails appearing in a sequence of $n$ coin flips, the largest domino snake that can be made using pieces up to $[n:n]$, and the longest trail on the complete graph $K_n$ with loops.
