Permutations with small maximal [formula omitted]-consecutive sums

Abstract

Let n and k be positive integers with n>k. Given a permutation (π1,…,πn) of integers 1,…,n, we consider k-consecutive sums of π, i.e., si≔∑j=0k−1πi+j for i=1,…,n, where we let πn+j=πj. What we want to do in this paper is to know the exact value of msum(n,k)≔minmax{si: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.