  We give a Littlewood-Richardson type rule for expanding the product of a
row-strict quasisymmetric Schur function and a symmetric Schur function in
terms of row-strict quasisymmetric Schur functions. We then discuss a family of
polynomials called Demazure atoms. We review the known characterizations of
these polynomials and then present two new characterizations. Finally, we
consider a family of polynomials called permuted basement nonsymmetric
Macdonald polynomials which are obtained by permuting the basement of the
combinatorial formula of Haglund, Haiman, and Loehr for nonsymmetric Macdonald
polynomials. We show that these permuted basement nonsymmetric Macdonald
polynomials are the simultaneous eigenfunctions of a family of commuting
operators in the double affine Hecke algebra.
