Motivated by Dohmen–Pönitz–Tittmann’s bivariate chromatic polynomial \(\chi _G(x,y)\), which counts all x-colorings of a graph G such that adjacent vertices get different colors if they are \(\le y\), we introduce a bivarate version of Stanley’s order polynomial, which counts order preserving maps from a given poset to a chain. Our results include decomposition formulas in terms of linear extensions, a combinatorial reciprocity theorem, and connections to bivariate chromatic polynomials.

中文翻译：

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二元阶多项式

由Dohmen–Pönitz–Tittmann的二元色多项式\（\ chi _G（x，y）\）激励，该函数对图G的所有x色进行计数，使得相邻顶点如果为（（\ le y \），则得到不同的颜色，我们引入了Stanley阶多项式的双变量版本，该序列对从给定的位姿到链的阶数保留映射进行计数。我们的结果包括根据线性扩展的分解公式，组合互易性定理以及与二元色多项式的连接。