加权键偏序集与一种新的色对称函数
Weighted bond posets and a new chromatic symmetric function
AI总结：
该研究利用加权键格引入图的新多项式与对称函数不变量，证明弦图下的多项式$γ$-正性与最高次齐次分量$e$-正性，并提出Schur-对数凹猜想。
AI中文摘要：
图的经典键格曾被用于Whitney的一个公式中以计算色多项式。本文中，我们使用键格的加权版本，引入并研究图的一种新多项式和一种新对称函数不变量。这种新多项式不变量的例子包括经典的Narayana多项式（对应路径图）、树-欧拉多项式（对应完全图）以及二项式-欧拉多项式（对应星图）。这些例子表明它与一般图-associahedra的$h$-多项式存在有趣的关联。我们证明，对于任意弦图，我们的多项式图不变量是$γ$-正的。\n 多重加权键偏序集产生了一种对称函数图不变量，它是色多项式的类似物。该对称函数的最高次齐次分量尤为值得关注。Haiman提出的停车函数对称函数是其中一个例子，第一作者结合多重括号李代数与着色外代数研究的对称函数也属于此类。上述$γ$-正性结果是最高次齐次分量的$e$-正性结果的一个特例，我们利用字典序壳化理论对任意弦图证明了该$e$-正性结果。我们猜想该对称函数是Schur-对数凹的，这一猜想可特例化为Huh的色多项式对数凹定理。
英文摘要：
The classical bond lattice of a graph was used in a formula of Whitney to compute the chromatic polynomial. In this paper, we use weighted versions of the bond lattice, to introduce and study a new polynomial and a new symmetric function invariant of a graph. Examples of the new polynomial invariant include the classical Narayana polynomials (for the path graph), the tree-Eulerian polynomials (for the complete graph), and the binomial-Eulerian polynomials (for the star graph). These examples suggest an interesting connection to $h$-polynomials of general graph-associahedra. We prove that for any chordal graph, our polynomial graph invariant is $γ$-positive. Multiweighted bond posets yield a symmetric function graph invariant that is an analog of the chromatic polynomial. The highest degree homogeneous component of this symmetric function is of particular interest. The parking function symmetric function introduced by Haiman arises as an example, as do symmetric functions studied by the first author in connection with multibracketed Lie algebras and with colored exterior algebras. The $γ$-positivity result mentioned above is a specialization of an $e$-positivity result for the highest degree homogeneous component, which we prove for any chordal graph using the theory of lexicographic shellability. We conjecture that this symmetric function is Schur-log-concave, which specializes to Huh's log-concavity theorem for the chromatic polynomial.
