For every non-identity conjugacy class in S_n, the set of attainable descent numbers is exactly the complete interval {1,2,...,M}, where M is the maximum descent count in that class.
Fulman,The Distribution of Descents in Fixed Conjugacy Classes of the Symmetric Groups, Journal of Combinatorial Theory, Series A, Volume 84, Issue 2, 1998, Pages 171-180 2
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The range of the des statistic for conjugacy classes in $S_n$
For every non-identity conjugacy class in S_n, the set of attainable descent numbers is exactly the complete interval {1,2,...,M}, where M is the maximum descent count in that class.