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We study Stirling permutations defined by Gessel and Stanley in [J. Combin. Theory Ser. A, 24 (1978), pp. 25-33]. We prove that their generating function according to the number of descents has real roots only. We use that fact to prove that the distribution of these descents and other equidistributed statistics on these objects converge to a normal distribution.

Real Zeros and Normal Distribution for Statistics on Stirling Permutations Defined by Gessel and Stanley