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. 

The distribution of the set of left-to-right maxima over 321-avoiding permutations with block number k is shown to be equal to 

the distribution of this set over  321-avoiding permutations with the last descent of the inverse permutation at position n - k.

This result is analogous to the classical 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.

Joint work with Ron Adin and Yuval Roichman.