The cardinality of s^{m-4}(S_{2m-4}) equals B_m plus (m^2 + 7m - 28)/2 for m at least 5.
A Proof of Julian West’s Conjecture That the Number of Two-Stacksortable Permutations of Length N Is 2(3n)!/((N + 1)!(2n + 1)!)
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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.