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3

Part of 1967 Czech and Slovak Olympiad III A

Problems(1)

Products of cyclic permutations

Source: Czech and Slovak Olympiad 1967, National Round, Problem 3

7/1/2024
Consider a table of cyclic permutations (n≥2n\ge2) 1,2,…,n−1,n2,3,…,n,1,⋮⋮⋱⋮⋮n,1,…,n−2,n−1. \begin{matrix} 1, & 2, & \ldots, & n-1, & n \\ 2, & 3, & \ldots, & n, & 1, \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n, & 1, & \ldots, & n-2, & n-1. \end{matrix} Then multiply each number of the first row by that number of the kk-th row that is in the same column. Sum all these products and denote sks_k the result (e.g. s2=1⋅2+2⋅3+⋯+(n−1)⋅n+n⋅1s_2=1\cdot2+2\cdot3+\cdots+(n-1)\cdot n+n\cdot1). a) Find a recursive relation for sks_k in terms of sk−1s_{k-1} and determine the explicit formula for sks_k. b) Determine both an index kk and the value of sks_k such that the sum sks_k is minimal.
Cyclicpermutationtablesquare table