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Sequences01:29

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Sequences are fundamental mathematical objects consisting of ordered lists of numbers that follow a specific rule or pattern. Sequences are critical in various mathematical concepts, including calculus, series, and number theory. They can model real-world phenomena such as population growth, financial investments, and physical processes like the diminishing height of a bouncing ball.Each number in a sequence is referred to as a term. Typically, the terms are denoted as a1, a2, a3,…, where...
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In systems where values diminish by a constant proportion at each stage, the resulting sequence follows a geometric structure. Each new value in the sequence is obtained by applying a fixed multiplier to the preceding term. This regular, proportional decline type is often used to represent processes involving gradual loss, such as energy dissipation or reduction in amplitude over time.When analyzing the total effect of such a process across unlimited iterations, the series of values is referred...
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Novel Sequence Discovery by Subtractive Genomics
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Number of longest increasing subsequences.

Phil Krabbe1, Hendrik Schawe1,2, Alexander K Hartmann1

  • 1Institut für Physik, Universität Oldenburg, 26111 Oldenburg, Germany.

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Summary

We analyzed the entropy of longest increasing subsequences (LISs) in random and i.i.d. sequences. The distribution is approximately Gaussian, with rare event tails deviating and a new rate function proposed.

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Area of Science:

  • Combinatorics and Probability Theory
  • Statistical Mechanics
  • Computer Science Algorithms

Background:

  • The study of longest increasing subsequences (LISs) is fundamental in combinatorics and algorithm analysis.
  • Understanding the statistical properties of LISs, such as their entropy, provides insights into sequence structures.
  • Previous research has focused on the length of LISs, with less attention paid to the distribution of the number of distinct LISs.

Purpose of the Study:

  • To investigate the statistical distribution of the entropy of longest increasing subsequences (LISs).
  • To analyze LIS entropy for two distinct sequence ensembles: random permutations and independent and identically distributed (i.i.d.) sequences.
  • To precisely characterize the probability distribution of LIS entropy, including extreme rare events.

Main Methods:

  • Development and application of sophisticated algorithms for exact counting of distinct LISs.
  • Computation of averages, variances, and detailed sampling of the probability distribution p(S) for LIS entropy.
  • High-precision numerical simulations to observe tail behaviors of the distribution.

Main Results:

  • The distribution of LIS entropy (S) is approximately Gaussian for both sequence ensembles.
  • Deviations from Gaussian behavior are observed in the far tails of the distribution, particularly for rare events (probabilities < 10^-600).
  • A novel large-deviation rate function was proposed and shown to accurately fit the observed data.

Conclusions:

  • The entropy of longest increasing subsequences exhibits near-Gaussian behavior with significant tail deviations.
  • The findings offer a deeper understanding of the statistical mechanics of sequences and their combinatorial properties.
  • The proposed rate function provides a valuable tool for analyzing extreme events in LIS entropy distributions.