Block decomposition of permutations and Schur-positivity

The block number of a permutation is the maximal number of components in its expression as a direct sum. We show that, for $321$-avoiding permutations, the set of left-to-right maxima has the same distribution when the block number is assumed to be $k$ as when the last descent of the inverse is assumed to be at position $n - k$. This result is analogous to the Foata-Sch\"utzenberger equi-distribution theorem, and implies that the quasi-symmetric generating function of descent set over $321$-avoiding permutations with a prescribed number of blocks is Schur-positive.