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A permutation is called dominant if its Lehmer code is a partition, or equivalently if it avoids the pattern $132$.

I can prove that given a permutation $v\in S_n$, there is a unique dominant permutation $\mu(v)$ such that $v\leq_R \mu(v)$ (where $\leq_R$ denotes right weak order) and if $\mu$ is any other dominant permutation such that $v\leq_R \mu$ then $\mu(v)\leq_R \mu$.

In particular, this implies that the meet of two dominant permutations in the weak order lattice is itself a dominant permutation.

I'm not really into the whole pattern avoidance aspect of permutations, nor have I much knowledge of what's going on in the study of weak order on Coxeter groups. Does anyone care that the meet of two $132$-avoiding permutations is itself a $132$-avoiding permutation?

In addition to this, dominant permutations end up being extremely important in Schubert calculus. Does anyone have any references that study their properties?

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    $\begingroup$ Is it not just the Tamari lattice you get this way? See in particular Theorem 9.6 of math.miami.edu/~wachs/papers/nonpure2.pdf. $\endgroup$
    – Sam Hopkins
    Commented Mar 11 at 17:52
  • $\begingroup$ @SamHopkins If the map in Proposition 9.10 is order-preserving, then yes, the dominant permutations form a lattice isomorphic to the Tamari lattice. I'm not sure that the join of two dominant permutations in weak order is dominant, though, so it may not be a sublattice of $S_n$ in weak order. I couldn't find a counterexample with a cursory search. $\endgroup$
    – Matt Samuel
    Commented Mar 11 at 18:09
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    $\begingroup$ This is me being lazy and not looking up all the details, but $x$ is $132$ avoiding if and only if $w_0 x$ is $312$-avoiding if and only if $w_0 x$ is $c$-sortable for $c = s_1 s_2 \cdots s_{n-1}$. (See Section 4 of arxiv.org/abs/math/0507186 .) The $c$-sortable elements form a sublattice (see Theorem 1.2 in arxiv.org/abs/math/0512339 ) and, for $c=s_1 s_2 \cdots s_{n-1}$ or $= s_{n-1} s_{n-2} \cdots s_2 s_1$, it is the Tamari lattice. $\endgroup$
    – David E Speyer
    Commented Mar 11 at 18:20
  • $\begingroup$ @DavidESpeyer Since $w_0$ reverses weak order, I guess that means the dominant permutations form a sublattice as well. Thanks. $\endgroup$
    – Matt Samuel
    Commented Mar 11 at 18:23
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    $\begingroup$ Yes, this should be the map that Reading calls $\pi_c^{\uparrow}$ (and the map to $312$-avoiding should be $\pi^c_{\downarrow}$). Both $\pi^{\uparrow}$ and $\pi_{\downarrow}$ are lattice homomorphisms. You basically want to read Reading's papers. $\endgroup$
    – David E Speyer
    Commented Mar 11 at 18:53

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