Asymptotics of self-overlapping permutations

TL;DR

This paper analyzes self-overlapping permutations within the theory of consecutive patterns, introducing a geometric interpretation and proving that any permutation can be written as a direct sum of non-self-overlapping blocks. It derives generating-function relations $S(z)$, $N(z)$, and $P(z)$, and shows that almost all permutations are non-self-overlapping, providing complete asymptotic expansions for the probability of self-overlapping with coefficients given by the non-self-overlapping counts $\mathfrak{n}_k$. It also develops asymptotic distributions for very tight pattern occurrences, offering explicit expansions and connecting these results to related permutation classes such as indecomposable and simple permutations, while outlining directions for extending the framework to graphs and other pattern families.

Abstract

In this work, we study the concept of self-overlapping permutations, which is related to the larger study of consecutive patterns in permutations. We show that this concept admits a simple and clear geometrical meaning, and prove that a permutation can be represented as a sequence of non-self-overlapping ones. The above structural decomposition allows us to obtain equations for the corresponding generating functions, as well as the complete asymptotic expansions for the probability that a large random permutation is (non-)self-overlapping. In particular, we show that almost all permutations are non-self-overlapping, and that the corresponding asymptotic expansion has the self-reference property: the involved coefficients count non-self-overlapping permutations once again. We also establish complete asymptotic expansions of the distributions of very tight non-self-overlapping patterns, and discuss the similarities of the non-self-overlapping permutations to other permutation building blocks, such as indecomposable and simple permutations, as well as their associated asymptotics.

Figures (7)

• Figure 1: Overlapping blocks in the permutation $214365$.
• Figure 2: Self-overlapping permutations of size at most $4$.
• Figure 3: Schema for reducing an overlapping range. If overlapping blocks have a non-empty intersection (gray areas on the figure), then this intersection can be slided to get smaller overlapping blocks (blue and violet areas).
• Figure 4: Decompositions of several self-overlapping permutations.
• Figure 5: Structure of self-overlapping permutations. If the overlapping block $\pi$ is a self-overlapping permutation itself, then its size is not the minimal overlapping range of the permutation $\sigma = \pi \oplus \tau \oplus \pi$.

Theorems & Definitions (22)

• Definition 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• Theorem 2.4
• proof
• Theorem 2.5
• proof