We consider words over the alphabet [k] = {1, 2, . . . , k}, k ≥ 2. For a fixed nonnegative integer p, a p-succession in a word w 1 w 2 . . . w n consists of two consecutive letters of the form (w i , w i + p), i = 1, 2, . . . , n − 1. We analyze words with respect to a given number of contained p-successions. First we find the mean and variance of the number of p-successions. We then determine the distribution of the number of p-successions in words of length n as n (and possibly k) tends to infinity; a simple instance of a phase transition (Gaussian-Poisson-degenerate) is encountered. Finally, we also investigate successions in compositions of integers.
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机译:我们考虑字母[k] = {1,2,...。 。 。 ,k},k≥2。对于固定的非负整数p,单词w 1 sub> w 2 sub>中的p继承。 。 。 w n sub>由两个连续的字母组成,形式为(w i sub>,w i sub> + p),i = 1、2,...。 。 。 ,n −1。我们分析包含给定数量的p继承的单词。首先,我们找到p成功数的均值和方差。然后,我们确定长度为n的单词中的p-接班人数量的分布,因为n(可能还有k)趋于无穷大;遇到一个简单的相变实例(高斯-泊松-简并)。最后,我们还研究了整数组成的连续性。
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