  We give a combinatorial formula for the inverses of the alternating sums of
free quasi-symmetric functions of the form F_{\omega(I)} where I runs over
compositions with parts in a prescribed set C. This proves in particular three
special cases (no restriction, even parts, and all parts equal to 2) which were
conjectured by B. C. V. Ung in [Proc. FPSAC'98, Toronto].
