Permutations with small maximal k-consecutive sums

Abstract

Let n and k be positive integers with n > k. Given a permutation (π 1 , . . . , πn) of integers 1, . . . , n, we consider k-consecutive sums of π, i.e., s i := k−1 j=0 π i+j for i = 1, . . . , n, where we let π n+j = π j . What we want to do in this paper is to know the exact value of msum(n, k) := min max{s i : i = 1, . . . , n} − k(n + 1) 2 : π ∈ Sn ,where Sn denotes the set of all permutations of 1, . . . , n. In this paper, we determine the exact values of msum(n, k) for some particular cases of n and k. As a corollary of the results, we obtain msum(n, 3), msum(n, 4) and msum(n, 6) for any n.