Partitioning permutations into monotone subsequences

Abstract

A permutation is $k$-coverable if it can be partitioned into $k$ monotone subsequences. Barber conjectured that, for any given permutation, if every subsequence of length $\left(\genfrac{}{}{0ex}{}{k+2}{2}\right)$ is $k$-coverable then the permutation itself is $k$-coverable. This conjecture, if true, would be best possible. Our aim in this paper is to disprove this conjecture for all $k\ge 3$. In fact, we show that for any $k$ there are permutations such that every subsequence of length at most $(k/6{)}^{2.46}$ is $k$-coverable while the permutation itself is not.

arXiv comment11 pages