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arXiv · 1909.11027

Characterization of quasirandom permutations by a pattern sum

Timothy F. N. Chan·Daniel Kral·Jonathan A. Noel·Yanitsa Pehova·Maryam Sharifzadeh·Jan Volec
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Abstract

It is known that a sequence Pi_i of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Pi_i converges to 1/24. We show that there is a set S of 4-point permutations such that the sum of the pattern densities of the permutations from S in the permutations Pi_i converges to |S|/24 if and only if the sequence is quasirandom. Moreover, we are able to completely characterize the sets S with this property. In particular, there are exactly ten such sets, the smallest of which has cardinality eight.

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Timothy F. N. Chan, Daniel Kral, Jonathan A. Noel, Yanitsa Pehova, Maryam Sharifzadeh, Jan Volec. 2022-07-15. Characterization of quasirandom permutations by a pattern sum. https://arxiv.org/abs/1909.11027

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