THE q-SERIES IN COMBINATORICS; PERMUTATION STATISTICS (Preliminary version) May 5, 2011 Dominique Foata and Guo-Niu Han Dominique Foata Guo-Niu Han D´epartement de math´ematique I.R.M.A. UMR 7501 Universit´eLouis Pasteur Universit´eLouis Pasteur et CNRS 7, rue Ren´e-Descartes 7, rue Ren´e-Descartes F-67084 Strasbourg, France F-67084 Strasbourg, France [email protected] [email protected] c These Lecture Notes cannot be reproduced without permission of the authors. Please, send remarks and corrections to: [email protected] or [email protected]. 1 Table of Contents Introduction 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 11.1. How to construct a bijection 11.2. The binary case 11.3. From the binary to the general case 11.4. Further properties of the transformation 11.5. Applications to permutations 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 3 D. FOATA AND G.-N. HAN 16. The Cauchy Identity 17. The Combinatorial definition of the Schur Functions 18. The inverse ligne 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 20.4. The t, q-maj extension for signed permutations 20.5. A first inversion number for signed permutations 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 D´esarm´enien Verfahren 23.3. Congruences for 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 Examples and Exercises Answers to the Exercises Notes References 4 Introduction The inverse Laplace transform maps each formal power series a(n)un n≥0 in one variable u into another series (a(n)/n!)un, whose coefficient of n≥0 order n is normalized by the factor n! We then obtain the so-called expo- nential generating function for the sequence (a(n)) (n 0). The normal- ization has numerous advantages: the derivative is obtaine≥ d by a simple shift of the coefficients; the exponential of a series can be explicitly calcu- lated; there are closed formulas for the exponential generating functions for several classical orthogonal polynomials, ... However the algebra of expo- nential generating functions cannot be regarded as the universal panacea. Further kinds of series are needed, for instance to express some generating series for the symmetric groups by certain statistics. In the middle of the eighteenth century Heine introduced a new class of series in which the normalized factor n! was replaced by a polynomial of degree n in another variable, more precisely, the series where the coefficient of order n is normalized by the polynomial denoted by (q; q)n, in another variable q, defined by 1, if n = 0; (0.1) (q; q) := n (1 q)(1 q2) (1 qn), if n 1. − − ··· − ≥ The algebra of those series has been largely developed by Jackson in the beginning of the twentieth century. It has then fallen into disuse, except perhaps in the field of Partition Theory, but has vigorously come back in several mathematical domains, in particular in the theory of Quantum Groups and naturally in Combinatorics. Those series have been named q-series. They are simply formal power series in two variables, say, u and q, where the latter variable q, used for the normalization, plays a privileged role. Let Ω[[u, q]] denote the algebra of formal series in two variables u and q, with coefficients in a ring Ω. Each element of that algebra can be expressed as a series (0.2) a = a(n, m) un qm, n≥0,m≥0 5 D. FOATA AND G.-N. HAN where, for each ordered pair (n, m), the symbol a(n, m) belongs to Ω. Such a series can be seen as a series in one variable u, with coefficients in the ring Ω[[q]] of series in one variable q, i. e., (0.3) a = un a(n, m) qm . n≥0 m≥0 For each integer n 0 the expression (q; q)n is a polynomial in q, which is invertible in Ω[[q≥]], since its constant coefficient is 1. Thus the series a can be rewritten as un (0.4) a = b(n; q), (q; q)n n≥0 where b(n; q) is the formal series in the variable q (0.5) b(n; q):=(q; q) a(n, m) qm . n · m≥0 Each formal series a written in the form (0.4) is called a q-series. The n coefficient u /(q; q)n is then a formal series in the unique variable q. The purpose of this memoir is to give a basic description of the algebra of those series and describe the use that has been made of them in Combinatorics, in particular for expressing the generating functions for certain statistics defined on permutations, words, multipermutations, signed permutations and other finite structures. It has been customary to regroup all the techniques that have been developed under the name of Permutation Statistic Study, even though the group of permutations is not the only group structure involved. The statistics themselves can be uni- or several-variable, or even set-valued. As will be seen, the q-series enter into the picture in a very natural way. The q-binomial theorem, which is stated and proved in the first section, is the basic tool in q-Calculus. It opens the door to all the q-series identities and also gives rise to two expansions of the q-exponential, as a q-series itself, and also as an infinite product. The polynomial (q; q)n, defined above, is next studied in a combinatorial context. This leads to a discussion of the so-called Mahonian statistics, especially the Major Index and the Inversion Number that will play an essential role in this memoir. One of our goals, indeed, is to try to understand why the so-called natural q-analogs of various numbers or polynomials can be derived by means of either one of those statistics. The Major Index is strongly related to the combinatorial theory of the representation of finite groups, particularly when dealing with various tableaux (standard, semi-standard, ... ). The inversion number requires 6 INTRODUCTION other techniques, in particular the so-called q-iteration method, that will be used on several occasions. The q-binomial coefficients or Gaussian polynomials that appear in many identities on q-series are studied in several combinatorial environ- ments, as is done in section 4. With the study of the q-multinomial co- efficient we are led to introduce the statistic “number of inversions” for classes of permutations with repetitions. We prefer to use the term “rear- rangement” (of a given finite sequence) or “word.” This is the content of section 5. The MacMahon Verfahren, introduced in section 6, is the first tool that makes possible the transcription of properties of certain statistics defined on the symmetric group or on some classes of rearrangements to the algebra of q-series. As a first application, it is shown that the Major Index has the same distribution as the number of inversions on each class of rearrangements. A careful study of the MacMahon Verfahren serves to find a q-extension of the traditional Eulerian polynomials, namely the Euler-Mahonian poly- nomials Am(t, q), that appear to be generating polynomials for classes of rearrangements by the bivariable statistic (des, maj). The statistic “des” is the number of descents that has been studied in several combinatorial contexts and “maj” is the Major Index. As shown in section 8, there are four equivalent definitions of the Euler-Mahonian polynomials. The proofs of those equivalences are based on fundamental techniques in q-Calculus, finite difference and iterative methods. The insertion technique that looks so natural when dealing with univariable statistics on the symmetric group becomes intricate for several- variable statistics. A marked word technique is presented in section 9 and appears to be successful for deriving a recurrence relation for the Euler- Mahonian polynomials Am(t, q). When the class of rearrangements is reduced to the symmetric group, the Euler-Mahonian polynomials become the so-called q-maj-Eulerian maj polynomials An(t, q), as they form a first q-analog of the traditional Eulerian polynomials An(t) in one variable t.