Abstract
Call a permutation $k$-inflatable if the sequence of its tensor products with uniform random permutations of increasing lengths has uniform $k$-point pattern densities. Previous work has shown that nontrivial $k$-inflatable permutations do not exist for $k \geq 4$. In this paper, we derive a general formula for the limit densities of patterns in the sequence of tensor products of a fixed permutation with each permutation from a convergent sequence. By applying this result, we completely characterize $3$-inflatable permutations and find explicit examples of $3$-inflatable permutations with various lengths, including the shortest examples with length $17$.
Highlights
In a broad sense, an object is called quasirandom if, asymptotically, it has similar properties to random objects of the class it belongs to
For permutations in particular [Coo[04], CP08], several different definitions of randomness are equivalent to a single concept of a quasirandom permutation sequence
We decided to write this paper when we discovered through experimentation that the example of a 3-inflatable permutation of length 9 in [CP08] is wrong
Summary
An object is called quasirandom if, asymptotically, it has similar properties to random objects of the class it belongs to. The n-th term of this sequence is the tensor product of the base permutation with a permutation chosen uniformly at random from all elements of Sn. A permutation is called k-inflatable when this construction results in a convergent permutation sequence that has uniform densities of all k-patterns. A permutation is called k-inflatable when this construction results in a convergent permutation sequence that has uniform densities of all k-patterns This topic has been studied in the past. We decided to write this paper when we discovered through experimentation that the example of a 3-inflatable permutation of length 9 in [CP08] is wrong With their methods as motivation, our primary contribution is a general method of computing asymptotic pattern densities in the random inflations of permutations. We show that if two permutations are both 3-inflatable, their tensor product is 3-inflatable
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