The expansion of immaculate functions in the ribbon basis

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).