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Shellability of simplicial balls and spheres (simplicial complexes whose geometric realizations are homeomorphic to balls and spheres) has been studied quite extensively. There are many explicit examples of shellable and nonshellable balls and spheres in the Library of Triangulations and Simplicial Complex Library. On the other hand, it is well-known that a shellable pseudomanifold is either a ball (if it has boundary) or a sphere (if its boundary is empty).

I am now searching for examples of shellable simplicial complexes that are not pseudomanifolds with dimension greater than 2. I understand that such examples have to be homotopy equivalent to a wedge of spheres, but cannot find any further discussion or any interesting constructions in the literature. I would really appreciate it if anyone can provide me with examples or references.

Edit: Two classes of examples I can think of are shifted complexes and matroids (which are in fact vertex decomposable, so they are too nice for the problem I am working on). Are there any other notable ones?

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    $\begingroup$ There are a vast number of known partially ordered sets (posets) satisfying a combinatorial condition known as lexicographic shellability. The order complex of such a poset is shellable. For example, any finite upper semimodular lattice is lexicographically shellable. One entry into this subject is the paper by Björner and Wachs at jstor.org/stable/1999359 $\endgroup$
    – Richard Stanley
    Commented Jun 28 at 0:33
  • $\begingroup$ @RichardStanley Thank you for the response, Professor Stanley! I have looked into these order complexes. Do you have any insights on whether these are all vertex decomposable? Prof. Wachs mentions in these notes that there have not yet been parallel techiniques for vertex decomposability on order complexes. On the other hand, per this post, it seems like shellable but not vertex decomposable flag complexes are hard to come by? $\endgroup$
    – mashedcarrots
    Commented Jun 29 at 0:59

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