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 \(\tilde{C}\) $\tilde{C}$ . In particular, we give a combinatorial formula for the Koornwinder polynomials \(K_{\lambda } = K_{\lambda }(z_1,\dots ,z_N; a,b,c,d; q,t)\) $K_{\lambda }=K_{\lambda }(z_1,\cdots ,z_N;a,b,c,d;q,t)$ , where \(\lambda = (1,\dots ,1,0,\dots ,0)\) $\lambda =(1,\cdots ,1,0,\cdots ,0)$ . We also give combinatorial formulas for all “open boundary ASEP polynomials” \(F_{\mu }\) $F_{\mu }$ , where \(\mu \) $\mu$ is a composition in \(\{-\,1,0,1\}^N\) ${\{-1,0,1\}}^N$ ; these polynomials are related to the nonsymmetric Koornwinder polynomials \(E_{\mu }\) $E_{\mu }$ 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.