Decompositions of bivariate order polynomials

We study bivariate order polynomials on bicolored posets. Richard Stanley first introduced univariate order polynomials in 1970. Recently, Sandra Zuniga Ruiz extended his work to bicolored posets and took a look at strict order polynomials on those posets. We continue her work by exploring what happens with weak as well as strict order polynomials on bicolored posets. Our main theorem decomposes bivariate weak order polynomials into a sum of type-ic order polynomials. We also prove a reciprocity theorem between weak and strict order polynomials.