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I am looking into interesting subsets of permutations, and there are several classes of permutations which are named. For example, there are

  • Derangements,
  • Alternating,
  • Grassmann permutations (at most one descent),
  • Bi-Grassmannian, wher both $\sigma$ an $\sigma^{-1}$ are Grassmann
  • Vexillary permutations (avoids 2143),
  • Richardson permutations,
  • Wachs permutations, (has anyone actually counted these?!)
  • The even permutations, $A_n$.
  • Flattened permutations (obtained from some set partition by removing the 'bars')

I am sure there are many more subsets of permutations with special names, and I would like to get some references for these (I plan to list them at my web-site www.symmetricfunctions.com ).

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    $\begingroup$ Dominant = 132-avoiding; fully commutative = 321-avoiding $\endgroup$
    – Sam Hopkins
    Commented Dec 13, 2022 at 13:53
  • $\begingroup$ @SamHopkins Any references for this particular use of terminology? $\endgroup$
    – Per Alexandersson
    Commented Dec 13, 2022 at 14:00
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    $\begingroup$ These are standard terms in the Schubert calculus/Coxeter world. For example, check out the Billey-Jockusch-Stanley paper where they are explored. The article on permutations at FindStat (findstat.org/CollectionsDatabase/Permutations) also has a short list of permutation classes significant to Schubert calculus. $\endgroup$
    – Sam Hopkins
    Commented Dec 13, 2022 at 14:02
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    $\begingroup$ You might also be interested in Bridget Tenner's website devoted to pattern avoidance: math.depaul.edu/~bridget/patterns.html. If you click "view the entire database" you can see many classes of permutations defined by avoidance: e.g., stack sortable, fully commutative, dominant, vexillary, etc. $\endgroup$
    – Sam Hopkins
    Commented Dec 13, 2022 at 14:19
  • $\begingroup$ @ "has anyone actually counted those": Propositions 3.2 and 3.4 in the preprint you cited give bijections. $\endgroup$
    – darij grinberg
    Commented Dec 13, 2022 at 14:46

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Some classes from the index entry "permutation" of EC1: André, connected, indecomposable, reverse alternating, separable, SIF (stabilized-interval-free), simple, simsun, Sundaram, standard. Also down-up and zigzag are synonyms for alternating, and up-down for reverse alternating. EC2 has Baxter, reduced Baxter, deque-sortable, smooth, stack-sortable, and 2-stack sortable.

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