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n. pl. vex·il·lar·ies
1. A member of the oldest class of army veterans who served under a special standard in ancient Rome.
2. A standard-bearer.
Of or relating to a vexillum.

[Latin vexillārius, from vexillum, flag; see vexillum.]


a standard bearer.
See also: Flags
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A combinatorial proof of Theorem 1.2 has been also obtained in [17] 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 [1] and vexillary permutations.
Before making this property precise, consider the following result, where vexillary permutations are exactly those that avoid the pattern 2143.
In [10], 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.
Any vexillary 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 [8]).
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
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: