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We describe a combinatorial formula for the coefficients when the dual immaculate quasisymmetric functions are decomposed into Young quasisymmetric Schur functions. We prove this using an analogue of Schensted insertion. Using this result, we give necessary and sufficient conditions for a dual immaculate quasisymmetric function to be symmetric. Moreover, we show that the product of a Schur function and a dual immaculate quasisymmetric function expands positively in the Young quasisymmetric Schur basis. We also discuss the decomposition of the Young noncommutative Schur functions into the immaculate functions. Finally, we provide a Remmel-Whitney-style rule to generate the coefficients of the decomposition of the dual immaculates into the Young quasisymmetric Schurs algorithmically and an analogous rule for the decomposition of the dual bases.

International audience We describe a combinatorial formula for the coefficients when the dual immaculate quasisymmetric func- tions are decomposed into Young quasisymmetric Schur functions. We prove this using an analogue of Schensted insertion. We also provide a Remmel-Whitney style rule to generate these coefficients algorithmically.

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Let C[X, Y ] denote the ring of polynomials with complex coefficients in the variables X=[x 1 , ..., x n ] and Y=[ y 1 , ..., y n ], let S n denote the symmetric group of order n!, let C m denote the cyclic group C m =[e 2?ijÂm : 0 j m&1] of order m, let H k denote the subgroup of order k of C m , and let G n, m =C m " S n (the wreath product of C m with S n

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