Inversion of some series of free quasi-symmetric functions

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 [B.C.V. Ung, Combinatorial identities for series of quasi-symmetric functions, in: Proc. FPSAC’08, Toronto, 2008].