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Asymptotic enumeration in pattern avoidance and in the theory of set partitions and asymptotic uniformity.

机译：模式避免以及集合划分和渐近均匀性理论中的渐近枚举。

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摘要

We demonstrate asymptotic properties of some popular combinatorial objects, including a partial answer to an open problem posed by Michael Atkinson and a general result on conditions for the coincidence of asymptotic normality and uniformity.;For a permutation pi ∈ Sn written in one line notation as pi = pi1pi2...pi n, we say pi contains the pattern sigma ∈ Sm if there exists a subsequence pi1=pim such that for all 1 ≤ j; k ≤ m it holds that pijpik if and only if sigmaj sigma k. Denote by sn(tau,sigma) the number of n-permutations which do not contain either of the patterns sigma and tau. Letting sigma' denote the pattern (m + 1)sigma; we construct classes of patterns for which the limit supremum of sn(123... r, sigma)1/n agrees with the limit supremum of sn(123...r, sigma')1/n for several classes of patterns sigma. We also construct classes of permutations which avoid 123...r and contain many patterns.;Many combinatorial sequences are of the form (an,k) where n ranges over the non-negative integers N and, for each n, there exists m = m( n) such that k ranges from 1 to m. We call such a sequence a combinatorial distribution. Many combinatorial distributions, upon rescaling, approach in distribution the normal distribution as n grows to infinity, a phenomenon we call asymptotic normality. A combinatorial distribution is said to be asymptotically uniform if, for each positive integer q and each residue class modulo q, the sum of coefficients an,k with k ≡ r (mod q) approaches 1/q as n grows to infinity. We call this asymptotic uniformity. We prove that if the generating polynomials for a combinatorial distribution have real, nonnegative zeros, asymptotic normality implies asymptotic uniformity. We apply this result to several sequences from the literature.;Finally, we present original results on the zeros of the Bell polynomials which were rst attained in proving the asymptotic uniformity of the Stirling numbers of the second kind Sn,k.
机译：我们证明了一些流行的组合对象的渐近性质，包括对迈克尔·阿特金森提出的一个开放问题的部分答案以及在渐近正态性和均匀性一致的条件下的一般结果;对于用一行表示的置换pi∈Sn pi = pi1pi2 ... pi n，如果存在子序列pi1 = pim使得对于所有1≤j，则pi包含模式sigma∈Sm。 k≤m且当且仅当sigmaj

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  • 作者

    Coleman, Micah Spencer.;

  • 作者单位

    University of Florida.;

  • 授予单位 University of Florida.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2008
  • 页码 91 p.
  • 总页数 91
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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