论文标题
塔玛里间隔和延长的战鱼之间的培训
A bijection between Tamari intervals and extended fighting fish
论文作者
论文摘要
我们将扩展的战鱼作为分支表面引入，也可以看作是通过简单重写规则定义的四分之一飞机的步行。我们在本文中提出的主要结果是直接两次培养的战斗鱼与塔玛里晶格的间隔之间交换了多个自然统计。该模型包括最近引入的战斗鱼（Duchi，Guerrini，Rinaldi，Schaeffer 2017），这些鱼被证明与同步的tamari间隔相等。使用对延长战鱼的双表面/步行观点，我们表明这些鱼的面积统计数据对应于塔玛里无连续性的距离统计数据（或链的最大长度）。我们还表明，根据较早的订单$ n^{5/4} $，统一随机扩展的尺寸尺寸的战鱼的平均面积，因此，tamari尺寸间隔的平均距离为订单$ n^{5/4} $。
We introduce extended fighting fish as branching surfaces that can also be seen as walks in the quarter plane defined by simple rewriting rules. The main result we present in the article is a direct bijection between extended fighting fish and intervals of the Tamari lattice that exchanges multiple natural statistics. The model includes the recently introduced fighting fish of (Duchi, Guerrini, Rinaldi, Schaeffer 2017) that were shown to be equinumerated with synchronized Tamari intervals. Using the dual surface/walk points of view on extended fighting fish, we show that the area statistics on these fish corresponds to the distance statistics (or maximal length of a chain) in Tamari invervals. We also show that the average area of a uniform random extended fighting fish of size $n$, and hence the average distance over the set of Tamari intervals of size $n$, is of order $n^{5/4}$, in accordance with earlier result for the subclass fighting fish.
