Parabolic Kazhdan-Lusztig R-polynomials for tight quotients of the symmetric groups

We give explicit closed combinatorial formulas for the parabolic Kazhdan-Lusztig R-polynomials of the tight quotients of the symmetric groups. We give two formulations of our result, one in terms of permutations and one in terms of Motzkin paths. As an application of our results we obtain explicit closed combinatorial formulas for certain sums and alternating sums of ordinary Kazhdan-Lusztig R-polynomials.