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 Sylvie Corteel Mathematics , 2006, Abstract: We present a simple a bijection between permutations of $\{1,..., n\}$ with $k$ descents and permutation tableaux of length $n$ with $k$ columns.
 Boaz Tsaban Mathematics , 2002, DOI: 10.1016/S0196-6774(03)00017-8 Abstract: A permutation P on {1,..,N} is a_fast_forward_permutation_ if for each m the computational complexity of evaluating P^m(x)$is small independently of m and x. Naor and Reingold constructed fast forward pseudorandom cycluses and involutions. By studying the evolution of permutation graphs, we prove that the number of queries needed to distinguish a random cyclus from a random permutation on {1,..,N} is Theta(N) if one does not use queries of the form P^m(x), but is only Theta(1) if one is allowed to make such queries. We construct fast forward permutations which are indistinguishable from random permutations even when queries of the form P^m(x) are allowed. This is done by introducing an efficient method to sample the cycle structure of a random permutation, which in turn solves an open problem of Naor and Reingold.  Axel Hultman Mathematics , 2013, Abstract: Given a permutation statistic$s : S_n \to \mathbb{R}$, define the mean statistic$\bar{s}$as the statistic which computes the mean of$s$over conjugacy classes. We describe a way to calculate the expected value of$s$on a product of$t$independently chosen elements from the uniform distribution on a union of conjugacy classes$\Gamma \subseteq S_n$. In order to apply the formula, one needs to express the class function$\bar{s}$as a linear combination of irreducible$S_n$-characters. We provide such expressions for several commonly studied permutation statistics, including the excedance number, inversion number, descent number, major index and$k$-cycle number. In particular, this leads to formulae for the expected values of said statistics.  Mathematics , 2013, Abstract: We propose several techniques to construct complete permutation polynomials of finite fields by virtue of complete permutations of subfields. In some special cases, any complete permutation polynomials over a finite field can be used to construct complete permutations of certain extension fields with these techniques. The results generalize some recent work of several authors.  Statistics , 2013, DOI: 10.1214/13-AOS1090 Abstract: Given independent samples from P and Q, two-sample permutation tests allow one to construct exact level tests when the null hypothesis is P=Q. On the other hand, when comparing or testing particular parameters$\theta$of P and Q, such as their means or medians, permutation tests need not be level$\alpha$, or even approximately level$\alpha$in large samples. Under very weak assumptions for comparing estimators, we provide a general test procedure whereby the asymptotic validity of the permutation test holds while retaining the exact rejection probability$\alpha$in finite samples when the underlying distributions are identical. The ideas are broadly applicable and special attention is given to the k-sample problem of comparing general parameters, whereby a permutation test is constructed which is exact level$\alpha$under the hypothesis of identical distributions, but has asymptotic rejection probability$\alpha\$ under the more general null hypothesis of equality of parameters. A Monte Carlo simulation study is performed as well. A quite general theory is possible based on a coupling construction, as well as a key contiguity argument for the multinomial and multivariate hypergeometric distributions.