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The Minimum Number of Monotone Subsequences

2002
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Electronic Journal of Combinatorics
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Erdős and Szekeres showed that any permutation of length $n \geq k^2+1$ contains a monotone subsequence of length $k+1$. A simple example shows that there need be no more than $(n \bmod k){{\lceil n/k \rceil}\choose {k+1}} + (k - (n \bmod k)){{\lfloor n/k \rfloor}\choose {k+1}}$ such subsequences; we conjecture that this is the minimum number of such subsequences. We prove this for $k=2$, with a complete characterisation of the extremal permutations. For $k > 2$ and $n \geq k(2k-1)$, we

doi:10.37236/1676
fatcat:y3rl5xf2kzdeffrk2lbzbqwf7m