Bivariate order polynomials

In the early 1900s, in an attempt to prove the 4-coloring theorem Birkhoff studied the number of ways to color a graph, known as the chromatic polynomial. In 1970, Richard Stanley found a way to decompose the chromatic polynomial using order polynomials along with many other interesting results. In 2003, Klaus Dohmen, Andre Pnitz, and Peter Tittman introduced a two variable generalization of the chromatic polynomial. We define bivariate order polynomials of bicolored posets. Furthermore, we provide a way to compute the bivariate order polynomial of any given bicolored poset. We also present the decomposition of the bivariate chromatic polynomial using bivariate order polynomials.