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

TL;DR

This work resolves a central question about non-consecutive pattern counts in random permutations by proving a near-halving lower bound: the expected number of distinct non-consecutive patterns satisfies $\mathbb{E}(X_n) \ge 2^{n-1}(1+o(1))$. The authors deploy the Stein–Chen Poisson-approximation framework to analyze pattern-occurrence processes $U_{k,l}$ of length $k$, with mean $\lambda = \binom{n}{k}/k!$, and construct a coupling that controls the total-variation distance to Poisson(λ) via error terms $T_1$, $T_2$, and $T_3$. A detailed study of overlaps between $k$-sets, including isomorphic overlaps and a notion of friendly permutations, yields sharp bounds on these error terms, enabling a global bound after summing over plausible $k$-values and selecting a cutoff $k_0 \approx n/2$. The result demonstrates that random permutations pack non-consecutive patterns only to about half of the total possible $2^n$ patterns in expectation, contradicting a prior conjecture and highlighting the nuanced role of dependencies among pattern occurrences. Overall, the paper advances the probabilistic toolkit for pattern-avoidance and pattern-count problems in combinatorics, with implications for related subsequence counting questions.

Abstract

Let $π_n$ 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 $π_n$ 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.

Theorems & Definitions (8)

• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 3.1