  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.
