Let [ n { = {1, …, n } and let Ω n be the set of all mappings from [ n { to itself. Let f be a random uniform element of Ω n and let T( f ) and B( f ) denote, respectively, the least common multiple and the product of the length of the cycles of f . Harris proved in 1973 that T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for random uniform mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k ( r ) ≥ 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x k + a over the integers modulo p , with p ≡ 1 (mod k). We exhibit and discuss our numerical results on this heuristic.

中文翻译：

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具有限制原像大小的函数的迭代周期

让 [n{ = {1, ...,n} 并让 Ωn是来自 [ 的所有映射的集合n{对自己。让F是 Ω 的随机均匀元素n并让 T(F) 和 B(F) 分别表示最小公倍数和循环长度的乘积F. 哈里斯在 1973 年证明 T 在分布中收敛到标准正态分布，并且在 2011 年，施穆茨在所有映射上获得了 T 和 B 期望的对数的渐近估计n节点。我们获得了随机均匀映射的类似结果n=氪原像大小限制为一组 {0,k} 的节点，其中ķ=ķ(r) ≥ 2。这是由于使用这些映射类作为启发式模型来统计以下形式的多项式Xķ + a 在整数模上p， 和p≡ 1 (mod k)。我们展示并讨论了我们在这个启发式上的数值结果。