The cardinality of s^{m-4}(S_{2m-4}) equals B_m plus (m^2 + 7m - 28)/2 for m at least 5.
Sorting and Preimages of Pattern Classes
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A Characterization of the $2m-4$ Case of Highly Sorted Permutations
The cardinality of s^{m-4}(S_{2m-4}) equals B_m plus (m^2 + 7m - 28)/2 for m at least 5.