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.