Abstract
We consider here the algebra of noncommutative symmetric functions, and two of its bases, the ribbon basis and the immaculate basis. Although a simple combinatorial formula is known for expanding an element of the ribbon basis in the immaculate basis, the reverse is not known. Using a sign-reversing involution, we prove an analogue of the classical Jacobi–Trudi formula which is an expression for an immaculate function indexed by a rectangle in the ribbon basis. We generalize this result to immaculate functions indexed by products of rectangles, and use this to prove a combinatorial formula for immaculate functions indexed by a rectangle of the form (2n).
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