It is known that a sequence {Π i } i∈N of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Π 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 Π 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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It is known that a sequence {Π i } i∈N of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Π 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 Π 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.