Some Instances of Homomesy Among Ideals of Posets
Some Instances of Homomesy Among Ideals of Posets
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偏序集理想中同质性的一些例子
DOI：
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发表时间：
2014
影响因子：
0.7
通讯作者：
Shahrzad Haddadan
中科院分区：
文献类型：
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作者：
Shahrzad Haddadan
给定一个置换$ Au$定义在一组组合对象$S$上，以及一些统计量$f：S
八箭头mathbb{R}$，我们说三重长角S， Au，f
如果f在所有轨道上具有相同的沿着平均值，则角f表现出同调性 Au$ in $S$.这种现象由Panyushev（2007）观察到，后来由Propp和Roby（2013）研究，命名和扩展。Propp和Roby研究homomesy在一组秩序理想的产品的两个链，有两个众所周知的排列，rowmotion和促进，统计的大小秩序理想。在本文中，我们将他们的结果推广到广义rowmotion和促进，以及更广泛的一类统计的产品的两个链。此外，我们在其他简单描述的偏序集上也得到了类似的结果。我们相信，我们在这里建立的框架可以证明更广泛的偏序集类的序理想的homomesy结果是富有成效的。
Given a permutation $ au$ defined on a set of combinatorial objects $S$, together with some statistic $f:S
ightarrow mathbb{R}$, we say that the triple $langle S, au,f
angle$ exhibits homomesy if $f$ has the same average along all orbits of $ au$ in $S$. This phenomenon was observed by Panyushev (2007) and later studied, named and extended by Propp and Roby (2013). Propp and Roby studied homomesy in the set of order ideals in the product of two chains, with two well known permutations, rowmotion and promotion, the statistic being the size of the order ideal. In this paper we extend their results to generalized rowmotion and promotion, together with a wider class of statistics in the product of two chains. Moreover, we derive similar results in other simply described posets. We believe that the framework we set up here can be fruitful in demonstrating homomesy results in order ideals of broader classes of posets.