论文标题
色度对称函数的电子基础系数
e-basis Coefficients of Chromatic Symmetric Functions
论文作者
论文摘要
斯坦利（Stanley）的一个众所周知的结果表明，给定具有色度对称功能的图形$ g $扩展到基本对称函数的基础上，$ x_g = \ sumc_λe_λ$，$c_λ$ for $λ$ for $λ$ for $λ_1'= k $（与$λ$等同于$ k $ k $ g n and cy acy） $ k $ sinks。 但是，众所周知。 The sink sequence of an acyclic orientation of $G$ is a tuple $(s_1,\dots,s_k)$ such that $s_1$ is the number of sinks of the orientation, and recursively each $s_i$ with $i > 1$ is the number of sinks remaining after deleting the sinks contributing to $s_1,\dots,s_{i-1}$.同等地，下沉序列给出了由无环方向引起的POSET级别的顶点数量。 史丹利（Stanley）的鲜为人知的随访结果确定了某些情况，在某些情况下，我们可以找到$ e $ bubasis系数的总和，这些系数将带有给定的部分接收器序列提供$ g $的无环方向的数量。当本身本身的关注时，当$ g $的稳定性为$ 2 $时，这个结果也将作为$ x_g $的$ e $ potitivity的简单证明。 在本文中，我们证明了该后续结果的顶点加权概括，并猜想了一个更强的版本，该版本可以接受类似的组合解释，以换取更大的$ e $ e $ e $ e $ - 表现的色度对称函数。特别是，猜想公式将对系数的总和$c_λ$的总和，其规定值为$λ_1'$和$λ_2'$，对于任何未经常的无爪图（不一定是无与伦比的无效图），如Stanley-Sputbridge猜想的设置）。
A well-known result of Stanley's shows that given a graph $G$ with chromatic symmetric function expanded into the basis of elementary symmetric functions as $X_G = \sum c_λe_λ$, the sum of the coefficients $c_λ$ for $λ$ with $λ_1' = k$ (equivalently those $λ$ with exactly $k$ parts) is equal to the number of acyclic orientations of $G$ with exactly $k$ sinks. However, more is known. The sink sequence of an acyclic orientation of $G$ is a tuple $(s_1,\dots,s_k)$ such that $s_1$ is the number of sinks of the orientation, and recursively each $s_i$ with $i > 1$ is the number of sinks remaining after deleting the sinks contributing to $s_1,\dots,s_{i-1}$. Equivalently, the sink sequence gives the number of vertices at each level of the poset induced by the acyclic orientation. A lesser-known follow-up result of Stanley's determines certain cases in which we can find a sum of $e$-basis coefficients that gives the number of acyclic orientations of $G$ with a given partial sink sequence. Of interest in its own right, this result also admits as a corollary a simple proof of the $e$-positivity of $X_G$ when the stability number of $G$ is $2$. In this paper, we prove a vertex-weighted generalization of this follow-up result, and conjecture a stronger version that admits a similar combinatorial interpretation for a much larger set of $e$-coefficient sums of chromatic symmetric functions. In particular, the conjectured formula would give a combinatorial interpretation for the sum of the coefficients $c_λ$ with prescribed values of $λ_1'$ and $λ_2'$ for any unweighted claw-free graph (not necessarily an incomparability graph, as in the setting of the Stanley-Stembridge conjecture).
