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 C̃. In particular, we give a combinatorial formula for the Koornwinder polynomials K_λ = K_λ(z_1,…,z_N; a,b,c,d; q,t), where λ = (1,…,1,0,…,0). We also give combinatorial formulas for all “open boundary ASEP polynomials” F_μ, where μ is a composition in {-1,0,1}^N; these polynomials are related to the nonsymmetric Koornwinder polynomials E_μ 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.