## q-SERIES IN COMBINATORICS; PERMUTATION STATISTICS

• 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

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