On a problem of team hiring
Résumé
Given two positive integers n and k with k ≤ n, let X denote the family of all the k-subsets of [n] := {1, 2, ..., n}. In this paper, assuming that the k-subsets in X hold some order of ranks, we consider the expected number of left-to-right maxima in some sequential ordering of X , under a random permutation of [n]. In the case k = 1, it is well-known that the answer is the nth harmonic number Hn = 1 + 1/2 + ... + 1/n = ln(n) + O(1). For general k, the contribution of this paper is about E * lex (n, k) and E * col (n, k), the expected numbers of left-to-right maxima respectively in the lexicographical and colexicographical orderings of X , achieved when the k-subsets in X hold their respective worst orders of ranks (to make the two values as large as possible). We show that E * lex (n, k) = E * col (n, k), and give an exact formula (not in closed form) for them. For estimating them, we further show that when k' := min{k, n − k} is fixed and n is big enough, they are asymtotic to (ln(n))^k'/k'!.
The problem we consider here can be viewed as an extension to the assistant-hiring problem, presented in the textbook Introduction to Algorithms by Cormen, Leiserson, Rivest, and Stein, for introducing probabilistic analysis and randomized algorithms.
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