How to Construct High Barrycades

Source abstract

Given two positive integers height $h$ and order $n$, the barrycade construction problem asks for a set of $h$ permutations of the integers from $1$ to $n$ such that all the proper partial sums given by these permutations are pairwise distinct. The name barrycade was coined by Richard K. Guy and refers to Barry Cipra, who introduced this kind of arrangement problem. A simple calculation shows that a solution can only exist for $n \ge 2h-2$ and it is conjectured that there always exists a solution for every height $h \ge 2$ and order $n \ge 2h-2$. In this work, for every height $h \ge 1$, we present a construction of a barrycade of height $h$ and order $n = 2h+3$. We also present a randomized heuristic that allows us to find a barrycade of height $h$ and conjectured optimal order $n=2h-2$, for every height $2 \leq h \leq 50$. Thus, we confirm the conjectured optimal order for all heights up to $50$. We also consider a related corral construction problem, where the permutations define a cyclic arrangement. In this setting, for every height $h \ge 1$, we present a construction of a corral of height $h$ and order $n=2h$. A heuristic approach, similar to the one used for barrycades, allows us to find a corral of height $h$ and conjectured optimal order $n=2h-1$, for every height $1 \leq h \leq 50$. We confirm Tomoki Nakamigawa's conjecture on well-dispersed partitions of cyclic groups for the number of parts up to $20$.