嵌套量子Dyck路与nabla（s_lambda）
Nested quantum Dyck paths and nabla(s_lambda)
摘要：
我们猜想了任意Schur函数在Bergeron-Garsia nabla算子下的像的单项式展开的一个组合公式。该公式涉及嵌套标记的Dyck路径加权面积和一个合适的“对角反转”统计。我们的模型包括作为特殊情况下，许多以前的连接纳布拉运营商的量子晶格路径。逆Kostka矩阵的组合学导致我们提出的公式当q=1时的一个初等证明。我们还概述了一种可能的方法来证明所有现存的纳布拉定理，减少一切建设的显式集合的签署，加权对象的符号反转对合。
英文摘要：
We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable "diagonal inversion" statistic. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the inverse Kostka matrix leads to an elementary proof of our proposed formula when q=1. We also outline a possible approach for proving all the extant nabla conjectures that reduces everything to the construction of sign-reversing involutions on explicit collections of signed, weighted objects.
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