A combinatorial proof of Theorem 1.2 has been also obtained in  for the non-factorial case, as well as an analogous determinant formula for skew flagged Grothendieck polynomials, special cases of which arise as the Grothendieck polynomials associated to 321-avoiding permutations  and vexillary
Before making this property precise, consider the following result, where vexillary permutations are exactly those that avoid the pattern 2143.
In , we showed that if a permutation w contains a vexillary p-pattern, then there is a reduced word for w possessing a reduced word f for p as a factor, possibly with a fixed positive integer added to each letter in f.
Schubert polynomial [Y.sub.v](x; 0) is equal to the Demazure character [K.sub.v] of the same index, and thus possesses the same description in terms of tableaux satisfying a flag property (see ).
It is relatively easy to prove a special case of Theorem 2.1 when the reverse of w is vexillary
, i.e., when w avoids 3412.
These are the vexillary permutations, those avoiding the pattern 2143.
This, together with the finiteness of the Lascoux-Schutzenberger tree and Stanley's result that [F.sub.[upsilon]] is a Schur function exactly when [upsilon] is vexillary, imply that
How can we characterize the "affine vexillary
Moreover, if w is vexillary, i.e., 2143-avoiding, the tableau P is the same for all reduced factorisations ofw.
It turns out that the permutations associated to moon polyominoes are indeed vexillary: