Permutations with small maximal k-consecutive sums

Let n and k be positive integers with n > k . Given a permutation ( π 1 , … , π n ) of integers 1 , … , n , we consider k -consecutive sums of π , i.e., s i ≔ ∑ j = 0 k − 1 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 ∈ S n , where S n 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 .