A Lower Bound on the Expected Number of Distinct Patterns in a Random Permutation

TL;DR

This paper establishes a lower bound on the expected number of distinct non-consecutive patterns in a random permutation, showing it is at least half of the total possible patterns, countering a previous conjecture.

Contribution

The authors prove a lower bound of at least 2^{n-1}(1+o(1)) for the expected number of distinct non-consecutive patterns in a random permutation, refuting a prior conjecture.

Findings

Expected number of non-consecutive patterns is at least 2^{n-1}(1+o(1)).

Counterexample to the conjecture that the expected number is close to 2^n.

Random permutations pack consecutive patterns nearly perfectly.

Abstract

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. The authors of [2] showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ was $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$, exhibiting the fact that random permutations pack consecutive patterns near-perfectly. A conjecture was made in [11] that the same is true for non-consecutive patterns, i.e., that there are $2^n(1-o(1))$ distinct non-consecutive patterns expected in a random permutation. This conjecture is false, but, in this paper, we prove that a random permutation contains an expected number of at least $2^{n-1}(1+o(1))$ distinct permutations; this number is half of the range of the number of distinct permutations.