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 C ~ . 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.