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.
Cellini,Cyclic eulerian elements, European Journal of Combinatorics 19 (1998), 545-552
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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.