insight - Scientific Computing - # Pattern Avoiding Permutations

Expected Length of Longest Subsequences Following Specific Up/Down Patterns in 132-Avoiding Random Permutations

Q: How does the expected length of the longest subsequence change for permutations avoiding patterns other than 132?

The expected length of the longest subsequence exhibiting a repeated up/down pattern within permutations avoiding a pattern of length three can change depending on the specific pattern being avoided. Symmetry Operations: As highlighted in Corollary 1, the expected lengths for permutations avoiding patterns other than 132 can be directly derived from the results obtained for 132-avoiding permutations. This is achieved by employing symmetry operations on permutations: reversal, complementation, and reversal-complementation. These operations establish bijections between the set of 132-avoiding permutations and permutations avoiding 213, 231, and 312. Consequently, the asymptotic behavior of the expected lengths for these patterns directly corresponds to the 132-avoiding case. Bijection with 123-avoiding Permutations: For permutations avoiding 123 and 321, a specific bijection with 132-avoiding permutations is utilized. This bijection, based on the concept of left-to-right minima, preserves the number of occurrences of the repeated up/down patterns UUD (or UUUD) between consecutive left-to-right minima. This implies that the expected lengths for these patterns in 123-avoiding and 321-avoiding permutations also align with the results derived for 132-avoiding permutations. Pattern-Specific Behavior: It's crucial to note that while the paper focuses on specific patterns (UUD, UUUD), the expected length's asymptotic behavior can differ for other repeated up/down patterns and avoidance patterns. The provided techniques offer a framework, but the specific constants and potentially even the growth order (e.g., linear in n) might vary.

Q: Could there be alternative combinatorial arguments, besides generating functions, to derive the asymptotic behavior of these expected lengths?

While the paper utilizes generating functions effectively, exploring alternative combinatorial arguments is intriguing. Here are some potential avenues: Direct Bijections: For simpler patterns, constructing direct bijections between permutations with specific longest subsequence lengths and other combinatorial objects counted by simpler sequences might be possible. This would necessitate a deep understanding of the structure imposed by both the repeated up/down pattern and the avoided pattern. Recursive Structures: Exploiting recursive structures within pattern-avoiding permutations could lead to recurrence relations for the expected lengths. Solving these recurrences, potentially using techniques like generating functions or other combinatorial methods, could provide alternative proofs. Probabilistic Interpretations: Interpreting the expected length probabilistically, perhaps as the expected value of some carefully defined random variable on the set of pattern-avoiding permutations, might offer new insights. This could involve analyzing the probabilities of certain substructures arising. It's important to acknowledge that the intricate interplay between the repeated up/down pattern and the avoided pattern makes finding elegant combinatorial arguments challenging. Generating functions provide a structured approach, but alternative perspectives could unveil deeper combinatorial connections.

Core Concepts

This research paper investigates the expected length of longest subsequences adhering to specific up/down patterns (UUD and UUUD) within random permutations that avoid the pattern 132, ultimately determining their asymptotic behavior.

Abstract

Bibliographic Information: PINSKY, R. G. (2024). Longest Subsequence for Certain Repeated Up/Down Patterns in Random Permutations Avoiding a Pattern of Length Three. arXiv preprint arXiv:2411.11482v1.
Research Objective: The paper aims to determine the asymptotic behavior of the expected length of the longest subsequence repeating the patterns UUD and UUUD in a uniformly random permutation avoiding the pattern 132.
Methodology: The study employs generating functions to analyze the expected number of complete patterns (UUD and UUUD) within maximal subsequences of 132-avoiding permutations. By deriving and solving linear equations for these generating functions, the researchers obtain explicit expressions that reveal the asymptotic behavior of the desired expectations.
Key Findings: The research demonstrates that the expected value of the longest increasing subsequence following the pattern UUD is asymptotic to 3/7n, while the expected value for the pattern UUUD is asymptotic to 4/11n, where n is the length of the permutation.
Main Conclusions: The paper concludes that the expected length of longest subsequences following the UUD and UUUD patterns in 132-avoiding permutations grows linearly with the permutation length, with specific constants of proportionality (3/7 and 4/11 respectively). These findings contribute to the understanding of pattern avoidance in permutations and the asymptotic behavior of longest subsequences with specific properties.
Significance: This research enhances the understanding of pattern-avoiding permutations, a topic with connections to various mathematical fields, including combinatorics and probability. The explicit calculation of asymptotic behavior for specific up/down patterns contributes to the broader study of longest subsequences in different types of permutations.
Limitations and Future Research: The paper focuses on 132-avoiding permutations and two specific up/down patterns. Exploring similar questions for other pattern-avoiding permutations and more general up/down patterns could be a potential avenue for future research. Additionally, investigating higher-order moments, such as variance, and exploring limiting distributions for these longest subsequence lengths could provide further insights.

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Stats

The expected value of the longest increasing subsequence following the pattern UUD is asymptotic to 3/7n.
The expected value of the longest increasing subsequence following the pattern UUUD is asymptotic to 4/11n.
For the repeated pattern UD (alternating subsequences), the corresponding asymptotic behavior is 1/2n.

Quotes

Key Insights Distilled From

Longest subsequence for certain repeated up/down patterns in random permutations avoiding a pattern of length three

by Ross G. Pins... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2411.11482.pdf

Longest subsequence for certain repeated up/down patterns in random permutations avoiding a pattern of length three

Deeper Inquiries

How does the expected length of the longest subsequence change for permutations avoiding patterns other than 132?

The expected length of the longest subsequence exhibiting a repeated up/down pattern within permutations avoiding a pattern of length three can change depending on the specific pattern being avoided.

Symmetry Operations: As highlighted in Corollary 1, the expected lengths for permutations avoiding patterns other than 132 can be directly derived from the results obtained for 132-avoiding permutations. This is achieved by employing symmetry operations on permutations: reversal, complementation, and reversal-complementation. These operations establish bijections between the set of 132-avoiding permutations and permutations avoiding 213, 231, and 312. Consequently, the asymptotic behavior of the expected lengths for these patterns directly corresponds to the 132-avoiding case.

Bijection with 123-avoiding Permutations: For permutations avoiding 123 and 321, a specific bijection with 132-avoiding permutations is utilized. This bijection, based on the concept of left-to-right minima, preserves the number of occurrences of the repeated up/down patterns UUD (or UUUD) between consecutive left-to-right minima. This implies that the expected lengths for these patterns in 123-avoiding and 321-avoiding permutations also align with the results derived for 132-avoiding permutations.

Pattern-Specific Behavior: It's crucial to note that while the paper focuses on specific patterns (UUD, UUUD), the expected length's asymptotic behavior can differ for other repeated up/down patterns and avoidance patterns. The provided techniques offer a framework, but the specific constants and potentially even the growth order (e.g., linear in n) might vary.

Could there be alternative combinatorial arguments, besides generating functions, to derive the asymptotic behavior of these expected lengths?

While the paper utilizes generating functions effectively, exploring alternative combinatorial arguments is intriguing. Here are some potential avenues:

Direct Bijections: For simpler patterns, constructing direct bijections between permutations with specific longest subsequence lengths and other combinatorial objects counted by simpler sequences might be possible. This would necessitate a deep understanding of the structure imposed by both the repeated up/down pattern and the avoided pattern.

Recursive Structures: Exploiting recursive structures within pattern-avoiding permutations could lead to recurrence relations for the expected lengths. Solving these recurrences, potentially using techniques like generating functions or other combinatorial methods, could provide alternative proofs.

Probabilistic Interpretations:  Interpreting the expected length probabilistically, perhaps as the expected value of some carefully defined random variable on the set of pattern-avoiding permutations, might offer new insights. This could involve analyzing the probabilities of certain substructures arising.
It's important to acknowledge that the intricate interplay between the repeated up/down pattern and the avoided pattern makes finding elegant combinatorial arguments challenging. Generating functions provide a structured approach, but alternative perspectives could unveil deeper combinatorial connections.

What are the implications of these findings for analyzing real-world data sequences that exhibit pattern avoidance properties?

The findings in this paper have potential implications for analyzing real-world data sequences that demonstrate pattern avoidance properties, particularly in scenarios where understanding increasing or repeating patterns is crucial:

Anomaly Detection: In time series data, deviations from expected patterns can indicate anomalies. Knowing the expected length of specific patterns in data expected to avoid certain permutations (due to underlying constraints) can help identify unusual events. For instance, in financial markets, unexpected price movements deviating from typical patterns could be flagged using these results.

Data Compression: Pattern-avoiding permutations and their properties can be leveraged for data compression algorithms. By understanding the expected lengths of repeating patterns, more efficient encoding schemes can be developed, especially for data known to exhibit specific avoidance patterns.

Biological Sequence Analysis: DNA and protein sequences often exhibit constraints that can be modeled as pattern avoidance. Analyzing the lengths of repeating subsequences within these sequences, considering the expected lengths under specific avoidance patterns, could provide insights into biological function and evolutionary relationships.

Social Network Analysis: Social networks and other relational data can be represented as graphs, where certain substructures (like specific patterns) might be rare or forbidden. The findings could be adapted to analyze the prevalence of specific motifs in these networks, even when certain patterns are avoided.
It's essential to recognize that applying these theoretical findings to real-world data requires careful consideration of the specific context and potential limitations. The data might not perfectly adhere to the assumptions of uniform randomness or specific pattern avoidance. Nevertheless, these results offer a valuable theoretical foundation for developing more sophisticated data analysis techniques in various domains.