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Forcing quasirandomness with 4-point permutations
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A combinatorial object is said to be quasirandom if it exhibits certain properties that are typically seen in a truly random object of the same kind. It is known that a permutation is quasirandom if and only if the pattern density of each of the twenty-four 4-point permutations is close to 1/24, which is its expected value in a random permutation. In other words, the set of all twenty-four 4-point permutations is quasirandom-forcing. Moreover, it is known that there exist sets of eight 4-point permutations that are also quasirandom-forcing. Breaking the barrier of linear dependency of perturbation gradients, we show that every quasirandom-forcing set of 4-point permutations must have cardinality at least five.
Forward citations
Cited by 3 Pith papers
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Sidorenko property and forcing in regular tournaments
For nearly regular tournaments, a tournament has the Sidorenko property exactly when it is transitive or a blow-up of the cyclic triangle whose three parts are transitive.
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Fuzzy latin squares and balanced permutation pattern statistics
For sufficiently large n, ordinary latin squares span the whole space of fuzzy latin squares, and the paper nearly classifies four-term vanishing and six-term non-vanishing examples.
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Forcing Quasirandomness in a Regular Tournament
All tournaments on at most five vertices that force quasirandomness in nearly regular tournaments are classified: eleven force it and nine do not.
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