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Vexillary permutations are an important family of permutations in Schubert calculus. There are several definitions, for example that they avoid the pattern 2143. Recall the Lehmer code of a permutation $w \in S_n$ is sequence $(c_1(w),c_2(w),\dots,c_{n-1}(w))$ where $c_i(w) = \#\{j >i: w(j) < w(i)\}$. Associated to each vexillary permutation $w \in S_n$ is a shape $\lambda(w)$ obtained by sorting the Lehmer code into an integer partition and a flag $\phi(w)$ obtained by sorting the tuple $(e_1,\dots,e_{n-1})$ where $$ e_i = \max\{j:j \geq i, c_j \geq c_i \}.$$ Note $e_i$ is 0 if this set is empty. The best reference I know for facts about vexillary permutations and their flags is Macdonald's Notes on Schubert Polynomials, which is very difficult to acquire but can be found in pdf form online. Recently, I've had need for the following statement about flags of vexillary permutations, which is (1.42) in Macdonald:

Proposition: Let $w \in S_n$ be vexillary with shape $\mu = (p_1^{m_1},\dots,p_k^{m_k})$ with $p_1 > \dots > p_k$ (so $m_1$ copies of $p_1$ and so on)and flag $\psi = (f_1^{m_1},\dots,f_k^{m_k})$ (so $m_1$ copies of $f_1$ and so on). Also, let $\mu' = (q_1^{n_1},\dots,q_k^{n_k})$. Then $$\lambda(w_0 w w_0) = \mu' \quad \mbox{and} \quad \phi(w_0w w_0) = \left((n{-}f_k)^{n_1},\dots,(n{-}f_1)^{n_k}\right).$$

Unfortunately, Macdonald's argument is "For once we shall leave the proof to the reader."

Question: Is there a reference for this fact that includes a proof?

I couldn't find this in Wach's original paper on vexillary permutations and flags, nor could I find it in several later papers.

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    $\begingroup$ Small comment: you never seem to use the $p$'s and $q$'s in the question. Also, perhaps the characterization of vexillary permutations as those whose Rothe diagrams can be transformed into a Young diagram by some permutation of rows and columns is useful here? $\endgroup$
    – Sam Hopkins
    Commented Oct 11 at 1:39
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    $\begingroup$ I need the $p$'s and $q$'s to define the $m$'s and $n$'s. The characterization you describe is very helpful for explaining why the shape is transposed, but I wasn't able to see how it related to the flag. $\endgroup$
    – Zach H
    Commented Oct 11 at 3:36
  • $\begingroup$ But what is $w_0$? $\endgroup$
    – Ilya Bogdanov
    Commented Oct 11 at 8:08
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    $\begingroup$ @IlyaBogdanov this is standard notation for the "longest permutation" $w_0=n, n-1, \ldots, 1$ (in one-line notation). Hence $w \mapsto w_0 w w_0$ is "reverse complement" a.k.a. $180^\circ$ rotation of the permutation matrix of $w$. $\endgroup$
    – Sam Hopkins
    Commented Oct 11 at 15:54

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