Characterization of quasirandom permutations by a pattern sum

Abstract

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.

Lower bound on the size of a quasirandom forcing set of permutations

“…Lemma 15. There are no three permutations π 1 , π 2 , π 3 and non-zero real coefficients t 1 , t 2 , t 3 such that max i∈[m] {|π i |} > 3 and i∈ [3] t i P π i = 0.…”

“…Recall the definition of mirror gradient polynomials P π (α, β) = P π (1 − α, β). Note that whenever the equality i∈ [3] t i P π i = 0, (8) holds, it also holds that i∈ [3] t i P π i = 0.…”

“…Moreover, by Lemma 16, the order of π 1 is at least four. Further, set S is linearly dependent by Lemma 8; hence there exist reals t 1 , t 2 , and t 3 such that i∈ [3] t i P π i = 0. Lemma 15 together with Lemma 13 then implies that precisely two of the coefficients are non-zero and their corresponding permutations have order two.…”

“…Besides graphs, there are results on quasirandomness of many different types of combinatorial structures, in particular, tournaments [2,6,13,18], hypergraphs [4,15,16,19,22], set systems [5], groups [17], subsets of integers [7], and Latin squares [10]. In this paper, we are concerned with quasirandomness of permutations as studied in [3,11,23].…”

“…Inspecting the proof given in [23], Zhang [27] observed that there exists a 16-element forcing set of 4-permutations. Bergsma and Dassios [1] identified an 8-element forcing set of 4-permutations and Chan [3] found three additional 8-element forcing sets of 4-permutations. In fact, these four 8-element forcing sets S of 4-permutations satisfy an even stronger property, which is called -forcing, i.e., a sequence of permutations is quasirandom if and only if the limit of the sum of the pattern densities of permutations in S converges to |S|/24.…”