In this paper, we study corners in tree‐like and permutation tableaux. Tree‐like tableaux are in bijection with other combinatorial structures, including permutation tableaux, and have a connection to the partially asymmetric simple exclusion process (PASEP), an important model of an interacting particles system. In particular, in the context of tree‐like tableaux, a corner corresponds to a node occupied by a particle that could jump to the right while inner corners indicate a particle with an empty node to its left. Thus, the total number of corners represents the number of nodes at which PASEP can move, that is, the total current activity of the system. As the number of inner corners and regular corners is connected, we limit our discussion to just regular corners and show that asymptotically, the number of corners in a tableau of length n is normally distributed. Furthermore, since the number of corners in tree‐like tableaux is closely related to the number of corners in permutation tableaux, we will discuss the corners in the context of the latter tableaux. Finally, using analogous techniques, we prove a central limit theorem for the number of corners in symmetric tree‐like tableaux and type‐B permutation tableaux.

中文翻译：

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与部分不对称简单排除过程相关的桌面中拐角数的渐近正态性

在本文中，我们研究了树状和置换画面中的角。树状的场景与其他组合结构（包括置换场景）相互融合，并与部分非对称简单排除过程（PASEP）（一种相互作用的粒子系统的重要模型）相关。特别是在树状场景的背景下，一个角对应于一个粒子所占据的节点，该节点可能会跳到右侧，而内角表示一个粒子，其左侧具有一个空节点。因此，拐角总数代表PASEP可以移动的节点数，即系统的当前活动总数。由于内角和正角的数量是相互关联的，因此我们的讨论只限于正角，并且渐近地表明，长度为一画面的角的数量n是正态分布的。此外，由于树状画面中的转角数量与置换画面中的转角数量密切相关，因此我们将在后一种画面中对转角进行讨论。最后，使用类似的技术，我们证明了对称树状表格和B型置换表格中拐角数目的中心极限定理。