where [lambda]' is the transpose of the partition [lambda], often referred to as conjugate partition [9, 10, 18].
The endomorphism is an involution and [omega]([s.sub.[lambda]]) = [s.sub.[lambda]'], where [lambda]' is the transpose of the partition [lambda], often referred to as conjugate partition [9, 10, 18].
However, the [omega] operator can be used to collect the compositions which rearrange a given partition so that there exists a bijection between these collections and the collections corresponding to the conjugate partition. In particular, note that since [omega] sends a quasisymmetric Schur function to a row-strict quasisymmetric Schur function, a method for writing the quasisymmetric Schur functions in terms of their duals under [omega] and vice versa would allow us to interpret the indexing compositions in much the same way as we interpret the indexing partitions for Schur functions and their images under [omega].