MathDB
2021 Combo #7: function sets

Source:

May 30, 2021
functionCombo

Problem Statement

Let S={1,2,…,2021}S = \{1, 2, \dots , 2021\}, and let F\mathcal{F} denote the set of functions f:S→Sf : S \rightarrow S. For a function f∈F,f \in \mathcal{F}, let Tf={f2021(s):s∈S},T_f =\{f^{2021}(s) : s \in S\}, where f2021(s)f^{2021}(s) denotes f(f(⋯(f(s))⋯ ))f(f(\cdots(f(s))\cdots)) with 20212021 copies of ff. Compute the remainder when ∑f∈F∣Tf∣\sum_{f \in \mathcal{F}} |T_f| is divided by the prime 20172017, where the sum is over all functions ff in F\mathcal{F}.
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