科研之友
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摘要
Given a permutation pi chosen uniformly from S(n), we explore the joint distribution of pi(1) and the number of descents in pi. We obtain a formula for the number of permutations with Des(pi) = d and pi(1) = k, and use it to show that if Des(pi) is fixed at d, then the expected value of pi(1) is d 1. We go on to derive generating functions for the joint distribution, show that it is unimodal if viewed correctly, and show that when d is small the distribution of pi(1) among the permutations with d descents is approximately geometric. Applications to Stein's method and the Neggers-Stanley problem are presented.
- 出版日期2010-4
