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I notice there is a strong connection between shellability of simplicial complexes and bistellar flips on these complexes; in particular, adding in a new facet of a shelling induces a bistellar flip on the boundary.

Is it always the case that bistellar flips preserve shellability of a complex? In other words, if I apply a bistellar flip to a shellable simplicial complex, is the resulting simplicial complex also shellable? If so, how can this be seen easily?

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Obviously not, because there exist non-shellable combinatorial spheres, but any combinatorial $n$-sphere is bistellar-equivalent to the boundary of the $(n+1)$-simplex.

The observation you mentioned is also in the very end of Lickorish's paper

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