Partitioning Permutations into Monotone Subsequences

  • David Wärn


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 ${k+2 \choose 2}$ 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 $k \geqslant 3$. 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.

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