This unpublished article (collaboration welcome)

https://www.researchgate.net/publication/329399313_Finding_or_counting_all_shellings_of_a_simplicial_complex

presents a systematic approach to decide shellability, which goes beyond extending partial shellings (that can confirm but hardly disprove shellability). Furthermore, our method (unlike Moriyama's algorithm) does not need the face-numbers. Instead of the n! permutations of the n facets we deal with certain admissible chains in certain posets of cardinality at most n.2^n (which is << n!). The shellability status of some matroid and some chessboard complexes with up to 24 facets is determined, or redetermined. Moreover the total number of shellings can be calculated. For instance the simplicial complex of all trees of the complete graph K_4 has exactly 722965625856 shellings. The previously known lower bound to the number of shellings was 6!=720.

This unpublished article (collaboration welcome) presents a systematic approach to decide shellability, which goes beyond extending partial shellings (that can confirm but hardly disprove shellability). Furthermore, our method (unlike Moriyama's algorithm) does not need the face-numbers. Instead of the $n!$ permutations of the $n$ facets we deal with certain admissible chains in certain posets of cardinality at most $n \cdot 2^n$ (which is $<< n!$). The shellability status of some matroid and some chessboard complexes with up to 24 facets is determined, or redetermined. Moreover the total number of shellings can be calculated. For instance the simplicial complex of all trees of the complete graph $K_4$ has exactly 722965625856 shellings. The previously known lower bound to the number of shellings was $6!=720$.