Rhombic staircase tableaux and Koornwinder polynomials

In this article we give a combinatorial formula for a certain class of Koornwinder polynomials, also known as Macdonald polynomials of type $\tilde{C}$. In particular, we give a combinatorial formula for the Koornwinder polynomials $K_{\lambda }=K_{\lambda }(z_1,\dots ,z_N;a,b,c,d;q,t)$, where $\lambda =(1,\dots ,1,0,\dots ,0)$. We also give combinatorial formulas for all ``open boundary ASEP polynomials’’ $F_{\mu }$, where $\mu$ is a composition in $\{-1,0,1{\}}^N$; these polynomials are related to the nonsymmetric Koornwinder polynomials $E_{\mu }$ up to a triangular change of basis. Our formulas are in terms of rhombic staircase tableaux, certain tableaux that we introduced in previous work to give a formula for the stationary distribution of the two-species asymmetric simple exclusion process (ASEP) on a line with open boundaries.