A proof of the Square Paths Conjecture
A proof of the Square Paths Conjecture
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方形路径猜想的证明
DOI：
10.1016/j.jcta.2017.06.013
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Emily Sergel
中科院分区：
文献类型：
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作者：
Emily Sergel
由Garsia和Haiman(1996)提出的修正的Macdonald多项式具有许多惊人的组合性质。一类这样的性质涉及将Bergeron和Garsia(1999年)的相关∇算子应用于基本对称函数。这种类型的第一个发现是(最近证实的)Haglund，Haiman，Loehr，Remmel和Ulyanov(2005年)的Shuffle猜想，它将表达式∇e n与停车函数联系起来。在(2007)中，Loehr和Warrington猜测了∇p n的一个类似的表达式，即平方路径猜想。Haglund和Loehr(2005)引入了时间表的概念来列举停车功能，在每条对角线上都有一组固定的汽车。本文将Hicks(2013)中时间表的概念和相关结果推广到带标号的方路上。然后，我们应用我们的新结果来证明平方路猜想。
The modified Macdonald polynomials, introduced by Garsia and Haiman (1996), have many astounding combinatorial properties. One such class of properties involves applying the related∇ operator of Bergeron and Garsia (1999) to basic symmetric functions. The first discovery of this type was the (recently proven) Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov (2005), which relates the expression∇ e n to parking functions. In (2007), Loehr and Warrington conjectured a similar expression for∇ p n which is known as the Square Paths Conjecture. Haglund and Loehr (2005) introduced the notion of schedules to enumerate parking functions with a fixed set of cars in each diagonal. In this paper, we extend the notion of schedules and some related results of Hicks (2013) to labeled square paths. We then apply our new results to prove the Square Paths Conjecture.