Abstract
We give a combinatorial formula for the inverses of the alternating sums of free quasi-symmetric functions of the form F_{ω(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].
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