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Everything about Random permutation statistics

Concept in combinatorics

The statistics of random permutations, such as the cycle structure of a random permutation, are of fundamental importance in the analysis of algorithms, especially of sorting algorithms, which operate on random permutations. Suppose, for example, that we are using quickselect to select a random element of a random permutation. Quickselect will perform a partial sort on the array, as it partitions the array according to the pivot. Hence a permutation will be less disordered after quickselect has been performed. The amount of disorder that remains may be analysed with generating functions. These generating functions depend in a fundamental way on the generating functions of random permutation statistics. Hence it is of vital importance to compute these generating functions. Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Random permutation statisticsA 5-minute orientation
  2. Skim a recent paper: Variance of statistics defined on random permutationsInfluential work since 2022
  3. Pick up a classic book: Introduction to Probability and StatisticsThe most-reprinted book on the subject
  4. Read the landmark paper: Permutation statistics of products of random permutationsCited 5 times
  5. Go sideways: Stirling numbers and exponential generating functions in symbolic combinatoricsA closely related topic
  6. Go sideways: Generating function transformationA closely related topic

📖 Definitions

No dictionary entry for this exact phrase.

🗂️ The subject at a glance

  1. The fundamental relation
  2. Number of permutations that are involutions
  3. Number of permutations that are mth roots of unity
  4. Number of permutations of order exactly k
  5. Number of permutations that are derangements
  6. Order of a random permutation
  7. Derangements containing an even and an odd number of cycles
  8. One hundred prisoners
  9. A variation on the 100 prisoners problem (keys and boxes)
  10. Number of permutations containing m cycles
  11. Expected number of cycles of a given size m
  12. Moments of fixed points

🏆 Landmark papers (most cited)

🆕 Influential since 2022

📚 Books worth reading

Ranked by number of editions on Open Library, a rough measure of how lasting a book is.

🔗 Related topics