For every integer h≥ 2 and every integer n≥ 2h-2, does there exist a collection of h permutations pi_1,…,pi_h of 1,…,n such that all proper partial sums ∑_i=1^kpi_j(i), with 1≤ j≤ h and 1≤ k≤ n-1, are pairwise distinct? Equivalently, whenever (j,k)≠(j',k'), one has ∑_i=1^kpi_j(i)≠∑_i=1^k'pi_j'(i).

Barrycade optimal-order conjecture

For every integer $h\ge 2$ and every integer $n\ge 2h-2$ , does there exist a collection of $h$ permutations $\pi_1,\ldots,\pi_h$ of $\{1,\ldots,n\}$ such that all proper partial sums $\sum_{i=1}^{k}\pi_j(i)$ , with $1\le j\le h$ and $1\le k\le n-1$ , are pairwise distinct? Equivalently, whenever $(j,k)\ne(j',k')$ , one has $\sum_{i=1}^{k}\pi_j(i)\ne\sum_{i=1}^{k'}\pi_{j'}(i)$ .

References

Primary source

arXiv

Additional references

How to Construct High Barrycades — arXiv — Jakub Binięda, Michał Dębski, Grzegorz Gutowski, Mateusz Milewski

combinatorics computer-assisted-mathematics

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

A new paper checks the conjectured bound through height 50 and gives a near-optimal construction, but does not settle the conjecture for every height.

The conjecture asserts that the counting lower bound $n=2h-2$ is attainable for every $h\ge 2$ . No proposer or original date is identified in the retrieved sources.

September 2026 paper

Jakub Binięda, Michał Dębski, Grzegorz Gutowski, and Mateusz Milewski report direct verification at the conjectured order $n=2h-2$ for every $2\le h\le 50$ , plus a uniform construction of order $n=2h+3$ for every height. They also verify a related cyclic-partition conjecture through $20$ parts. The paper explicitly leaves the all-heights conjecture unresolved.

Current status (as of September 2026): the conjecture is verified through $h=50$ and has a uniform near-optimal construction, but attainment of $n=2h-2$ for all $h\ge 2$ remains open.

Sources

No solutions have been posted yet.