This volume contains the scientific contributions that old friends, colleagues, admirers, former students have made to George Andrews on the occasion of his sixtieth birthday. We are pleased and honored to welcome it, as number 42, in our series Seminaire Lotharingien de Combinatoire.
The idea of celebrating George Andrews in one of our Seminars was suggested by Peter Paule during our 40th meeting at the Domaine-Saint-Jacques in March 1998. All the participants loved the idea. It remained to materialize.
The meeting was to take place in the Fall 1998, preferably in a more exotic area than the Vosges mountains. Dr Vito Stigliani, the director of the Istituto Italiano di Cultura in Strasbourg, suggested his native province of Basilicata in Southern Italy. We then contacted Domenico Senato of the Università degli Studi della Basilicata at Potenza, who promised to help us organize the rencontre. And he did so, beyond all our expectations.
Without the efficiency of the Potenza team, Domenico Senato, Elvira Di Nardo and Paolo Vitolo and the financial support of the Regione Basilicata and the dipartimento di Matematica, at the Università degli Studi della Basilicata nothing would have been possible.
The meeting took place at Maratea, some 200 kilometers south of Naples, from August 31 to September 6, 1998. The participants were:
G.E. Andrews (University Park), V. Strehl, R. Koenig (Erlangen), P. Paule, A. Riese (Linz), J. Désarménien, H. Olivier, P.-A. Picon, A. Lascoux, D. Perrin, B. Gauthier, J.-F. Beraud (Marne-la-Vallée), D. Foata, G. Han, J.-P. Jouanolou (Strasbourg), H. Gaudier (Valenciennes), A.M. Garsia, D. Little (San Diego), D. Bressoud (Saint Paul), R. Askey (Madison), A. Carpi (Arco Felice), D. Senato, O. di Vincenzo, P. Vitolo, E. Di Nardo (Potenza), C. Krattenthaler (Wien), M. Mureddu (Cagliari), F. Brenti, A. de Luca, C. Malvenuto (Roma), P. Kirschenhofer (Leoben), J. Olsson (Copenhagen), K. Alladi (Gainesville), S. Milne (Columbus), A. Zvonkine (Bordeaux), H. Wilf (Philadelphia), A. Schilling, O. Warnaar (Amsterdam), F. Bergeron (Montreal), N. Bergeron (Toronto), J. Zeng (Lyon), M. Cerasoli (L'Aquila), A. Berkovich (Stony Brook), A. Brini (Bologna), L. Carini (Messina), M. Joswig (Berlin), A. Giambruno, A. Valenti (Palermo) A. Marini, R. Zizza (Milano), K. Simon (Zuerich).
Several colleagues and coauthors of George Andrews could not come to Maratea, but still wanted to send us their contributions. We are proud to include them in the present volume, which contains seventeen papers ranging from Classical Number Theory (the theory of partitions dear to George Andrews) to Classical and Algebraic Combinatorics.
We are most thankful to George Andrews for his original paper in which he describes how several great mathematicians, dead or living, have influenced his mathematical career.
The second paper could only have been written by his good friend, Richard Askey, who appears to be, in Doron Zeilberger's inimitable language, the "guru" of special functions, George himself being the "q-guru"! Richard very convincingly explains how the year that George spent at Madison in 1975-76 was decisive for them both.
There is no better homage to George and his familiarity with Ramanujan's works than this long memoir by Bruce C. Berndt and Ken Ono on Ramanujan's Unpublished Manuscript and the Tau Functions. Bruce has spent a great part of his career deciphering the manuscripts of the great Indian number theorist. In his contribution jointly written with Ken he presents Ramanujan's complete manuscript, providing details and an extensive commentary.
Will the marvelous savoir-faire of classical analysts dealing with special functions be replaced by advanced computer algebra softwares in the future? Part of it, for sure. For the time being we welcome those new maple packages prepared by professional mathematicians, such as Frank Garvan, which are of a great help in dealing with q-series calculations.
In the subsequent paper G.-N. Han, A. Randrianarivony and J. Zeng have imagined a new class of q-secant and q-tangent numbers whose ordinary generating functions have simple continued fraction expansions. Those new q-numbers also have fascinating combinatorial properties. We most welcome this contribution to Combinatorial Number Theory, for it may help us fill the gap existing between geometric interpretations derived from the algebra of exponential series and those derived from the study of continued fractions.
The simple proofs of several results on representations of numbers by quadratic forms obtained by Mike Hirschhorn show his great knowledge of that part of Classical Number Theory that goes back to Dirichlet and Lorenz.
Dongsu Kim and Dennis Stanton calculate the generating function for words with several simultaneous weights, all derived from the celebrated major index statistic. Their new results can be immediately applied to integer partition theory.
The extension of Franklin's bijection found by David P. Little provides a very convincing explanation for the appearances of certain terms in the expansion of the pentagonal number product, when finitely many terms are deleted in the beginning.
George E. Andrews and Peter Paule revisit the works of MacMahon, particularly his computational method for solving problems in connection with linear homogeneous diophantine inequalities and equations. They show that Partition Analysis can be also used for proving hypergeometric multisum identities that arise in physics and algebraic combinatorics.
The generalized Kostka polynomials are q-analogues of the multiplicities of a specific finite-dimensional irreducible representation of the linear group. Their combinatorial properties are difficult to handle. Anatol N. Kirillov, Anne Schilling and Mark Shimozono review several representations of those polynomials, such as the charge, path space, quasi-particle and bosonic representation. They also describe a bijection between Littlewood-Richardson tableaux and rigged configurations.
The Jacobi triple product identity has fascinated many generations of mathematicians. In a short note Herbert S. Wilf finds a new and surprising number-theoretic content of that classical identity.
The paper by Arturo Carpi and Aldo de Luca on Words and Repeated Factors belongs to a long research tradition that consists of
studying the unavoidable regularities of words of arbitrary large
length. Here they characterize the sets of factors of a finite word that make possible the reconstruction of the entire word.
The Californian School, duly represented by Adriano Garsia, Mark Haiman and Glenn Tesler has made an extensive study of the Macdonald polynomials in the past ten years. They are kind enough to send us a very significant part of their monumental project that contains explicit plethystic formulas for the Macdonald q,t-Kostka coefficients.
The survey proposed by S. Ole Warnaar on supernomial coefficients, Bailey's lemma and Rogers-Ramanujan-type identities indicates that the famous Bailey lemma now has an A(2)-extension that makes possible the derivation of natural analogues of the classical A(2)-Rogers-Ramanujan identities. The paper also gives the description of unsolved problems, such as the supernomial inversion, the A(n-1) and higher level extensions, as well as several conjectures involving A(n-1) basic hypergeometric series, string functions and cylindric partitions.
It is still a mystery of mathematical tradition to see the triple and quintuple product identities proved and reproved periodically. The purpose of D. Foata and G.-N. Han, on the occasion of the discovery of a septuple product identity imagined and proved by Farkas and Kra, is to show that all those identities, including the brand new one, can be derived by using purely "manipulatorics" methods, and probably by computer algebra softwares in the future.
In the square-ice model Korepin et al. have derived a partition function that is symmetric in two sets of variables. What is not clear is how to separate the variables. Alain Lascoux, with his efficient method of divided differences obtains the desired symmetric function as a product of two rectangular matrices, each of them involving only one set of variables.
In the past five years Christian Krattenthaler has obtained several precise evaluations of various countings, in particular in tiling. Most of those countings were first expressed in terms of determinants. The greatest difficulty was to calculate them. In his paper Christian unveils the clues of his fabulous calculating techniques.
All in all, it has been a rewarding experience to have Lotharingia meet Pennsylvania in the person of George Andrews on the shores of old Mediterranean.