Reconstructing compositions

We consider the problem of reconstructing compositions of an integer from their subcompositions, which was raised by Raykova (albeit disguised as a question about layered permutations). We show that every composition ww of n⩾3k+1n⩾3k+1 can be reconstructed from its set of kk-deletions, i.e., the set of all compositions of n-kn-k contained in ww. As there are compositions of 3k3k with the same set of kk-deletions, this result is best possible.