Distribution of Descents in Matchings
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The distribution of descents in certain conjugacy classes of \(S_n\) has been previously studied, and it is shown that its moments have interesting properties. This paper provides a bijective proof of the symmetry of the descents and major indices of matchings (also known as fixed point free involutions) and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings.
KeywordsEnumerative combinatorics Probabilistic combinatorics Central limit theorem
Mathematics Subject ClassificationPrimary 05A19 60F05 Secondary 60C05 62E20
The author would like to thank Brendon Rhoades for introducing the author to the world of Young tableaux and helping come up with the bijection in Sect. 3. The author would also like to thank Sangchul Lee for the suggestion of considering moment generating functions instead of characteristic functions and helping with the calculations of bounding the integrals in Sect. 4. Finally, the author would like to thank Jason Fulman for the suggestion of the problem and his guidance.
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