June 24, 2004. 207 pages

Dominique Foata, Guo-Niu Han

q-SERIES IN COMBINATORICS; PERMUTATION STATISTICS
(preliminary version; to be updated shortly)

• pdf version (1400 K)

• 1. The q-binomial theorem
• 2. Mahonian Statistics
  • 2.1. The inv-coding
  • 2.2. The maj-coding
  • 2.3. The den-coding
• 3. The algebra of q-binomial coefficients
• 4. The q-binomial structures
  • 4.1. Partitions of integers
  • 4.2. Nondecreasing sequences of integers
  • 4.3. Binary words.
  • 4.4. Ordered Partitions into two blocks
• 5. The q-multinomial coefficients
• 6. The MacMahon Verfahren
• 7. A refinement of the MacMahon Verfahren
• 8. The Euler-Mahonian polynomials
  • 8.1. A finite difference q-calculus
  • 8.2. A q-iteration method
• 9. The Insertion technique
• 10. The two classes of q-Eulerian polynomials
• 11. Major Index and Inversion Number
• 12. Major and Inverse Major Indices
  • 12.1. The biword expansion
  • 12.2. Another application of the MacMahon Verfahren
• 13. A four-variable distribution
• 14. Symmetric Functions
  • 14.1. Partitions of integers
  • 14.2. The algebra of symmetric functions
  • 14.3. The classical bases
• 15. The Schur Functions
• 16. The Cauchy Identity
• 17. The Combinatorial definition of the Schur Functions
• 18. The inverse ligne of route of a standard tableau
• 19. The Robinson-Schensted correspondence
  • 19.1. The Schensted-Knuth algorithm
  • 19.2. A combinatorial proof of the Cauchy identity
  • 19.3. Geometric properties of the correspondence
  • 19.4. A permutation statistic distribution
• 20. Eulerian Calculus; the first extensions
  • 20.1. The signed permutations
  • 20.2. Pairs of permutations
  • 20.3. The q-extension
• 21. Eulerian Calculus; the analytic choice
  • 21.1. Inversions for signed permutations
  • 21.2. Basic Bessel Functions
  • 21.3. The iterative method
• 22. Eulerian Calculus; finite analogs of Bessel functions
  • 22.1. Signed biwords
  • 22.2. Signed bipermutations
  • 22.3. Signed biwords and compatible bipermutations
  • 22.4. The last specializations
• 23. Eulerian Calculus; multi-indexed polynomials
  • 23.1. The bi-indexed Eulerian polynomials
  • 23.2. The Desarmenien Verfahren
  • 23.3. Congruences of bi-indexed polynomials
  • 23.4. The signed Eulerian Numbers
• 24. The basic and bibasic trigonometric Functions
  • 24.1 The basic and bibasic tangent and secant functions
  • 24.2. Alternating permutations
  • 24.3. Combinatorics of the bibasic secant and tangent
• Exercices and examples
• Answers to the exercices
• Notes