The minimum Manhattan distance and minimum jump of permutations

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Abstract

Let π be a permutation of {1,2,,n}. If we identify a permutation with its graph, namely the set of n dots at positions (i,π(i)), it is natural to consider the minimum L1 (Manhattan) distance, d(π), between any pair of dots. The paper computes the expected value (and higher moments) of d(π) when n and π is chosen uniformly, and settles a conjecture of Bevan, Homberger and Tenner (motivated by permutation patterns), showing that when d is fixed and n, the probability that d(π)d+2 tends to ed2d.

The minimum jump mj(π) of π, defined by mj(π)=min1in1|π(i+1)π(i)|, is another natural measure in this context. The paper computes the asymptotic moments of mj(π), and the asymptotic probability that mj(π)d+1 for any constant d.

Keywords

Permutations
Asymptotic enumeration
Manhattan distance

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This author is supported by NSF grants DMS-1162172 and DMS-1600116.