On the outstanding elements of permutations
نویسنده
چکیده
If σ is a permutation of n letters, say that j is an outstanding element (Fr. élément saillant) of σ if whenever i < j we have σ(i) < σ(j). A famous theorem of A. Rényi [R] (see also [C]) states that the number of permutations of n letters that have k outstanding elements is equal to the number that have k cycles, and therefore there are [ n k ] such permutations, the latter being the unsigned Stirling number of the first kind. Here we investigate other properties of the outstanding elements, and show the following: (I) The average of the rth outstanding elements, over all permutations of n letters that have that many, is ∼ (log n)r−1/(r − 1)!, for fixed r and n→∞. (II) The average value of a permutation σ at its rth outstanding element, among all permutations that have that many, is ∼ (1− 1 2r )n (n→∞).
منابع مشابه
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