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Asymptotics of commuting -tuples in symmetric groups and log-concavity
Kathrin Bringmann1, Johann Franke1, Bernhard Heim1
1Division of Mathematics, Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany.
Research in Number Theory
|October 7, 2024
Summary
This study derives asymptotic formulas for the number of commuting tuples in symmetric groups. It also establishes log-concavity criteria and a Bessenrodt-Ono type theorem for specific sequences.
Area of Science:
- Group theory
- Combinatorics
- Number theory
Background:
- The study of commuting elements within algebraic structures like symmetric groups is a key area in group theory.
- Understanding the distribution and properties of such elements provides insights into the group's structure.
Purpose of the Study:
- To derive asymptotic formulas for the number of commuting tuples in symmetric groups.
- To establish general criteria for log-concavity applicable to these counts.
- To present a Bessenrodt-Ono type theorem for related sequences.
Main Methods:
- Asymptotic analysis to approximate the number of commuting tuples.
- Development of general criteria for proving log-concavity.
- Algebraic number theory techniques to establish inequalities.
Main Results:
- Asymptotic formulas for the number of commuting tuples in symmetric groups.
- Demonstration of log-concavity for these counts using derived criteria.
- A Bessenrodt-Ono type inequality for specific sequences c(n).
Conclusions:
- The research provides precise asymptotic behavior for commuting tuples in symmetric groups.
- The established log-concavity criteria offer a powerful tool for analyzing related combinatorial sequences.
- The Bessenrodt-Ono type theorem extends existing results in additive number theory.
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