The organ pipe permutation

Let $(p_1 ,p_2 , \cdots ,p_n )$ be a probability distribution, $\pi = (\pi _1 ,\pi _2 , \cdots ,\pi _n )$ be a permutation of $1,2, \cdots ,n$ and $X_1 ,X_2 , \cdots ,X_k $ be $k$ independent and identically distributed random variables with distribution $P(X = i) = p\pi _i $. It is known that the organ pipe permutation $\pi ^ * $ makes the range \[ D(k,\pi ) = \max\limits_{1 \leqq j \leqq k} X_i - \min\limits_{1 \leqq j \leqq k} X_i \]a stochastic minimum for $k = 2$ (P. P. Bergmans, Information and Control, 20 (1972), pp. 331-350), and minimal on the average for general $k$ (J. R. Bitner and C. K. Wong, 8 (1979), pp. 479-498). We prove the stochastic minimality for general $k$ and study a natural extension of the organ pipe permutation that is optimal when certain constraints are placed on the possible choices of $\pi $.