By Dennis Stanton (es.)

This IMA quantity in arithmetic and its purposes q-Series and walls is predicated at the complaints of a workshop which used to be an essential component of the 1987-88 IMA software on utilized COMBINATORICS. we're thankful to the medical Committee: Victor Klee (Chairman), Daniel Kleitman, Dijen Ray-Chaudhuri and Dennis Stanton for making plans and imposing a thrilling and stimulating 12 months­ lengthy application. We specifically thank the Workshop Organizer, Dennis Stanton, for organizing a workshop which introduced jointly a few of the significant figures in numerous examine fields during which q-series and walls are used. A vner Friedman Willard Miller, Jr. PREFACE This quantity includes the court cases of the Workshop on q-Series and Parti­ tions held on the IMA on March 7-11, 1988. additionally incorporated are papers via Goodman and O'Hara, Macdonald, and Zeilberger on unimodality. This paintings was once of substan­ tial curiosity and mentioned by way of many contributors within the Workshop. The papers were grouped into 4 components: identities, unimodality of Gaus­ sian polynomials, consistent time period difficulties and similar integrals, and orthogonal polynomials. They symbolize a move element of the hot paintings on q-series includ­ ing: walls, combinatorics, Lie algebras, research, and mathematical physics. i need to thank the employees of the IMA, and its administrators, Avner Friedman and Willard Miller, Jr., for supplying an excellent setting for the Workshop. Patricia Brick and Kaye Smith ready the manuscripts.

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A formal proof of these identities will involve writing all the integers as a combination of powers of 10, invoking the associative and distributive laws, and then using the 100 "lemmas" that comprise the multiplication table and the 100 lemmas that constitute the addition table. Since arithmetic with specific integers is so much part of our culture, it would not occur to anyone to give a proof of a statement like 2), and the proof is considered routine by general consensus. + 3: Way back in 1202, when Liber Abaci first appeared, identity 3a) was a deep theorem, and there were very few people who understood its proof.

A, to appear. [8] D. ), q-Series and Partitions, IMA Volumes in Mathematics and its Applications, Springer-Verlag, New York (1989). o ON THE GAUSSIAN POLYNOMIALS· FREDERICK M. GOODMANt AND KATHLEEN M. O'HARAt The main features of the constructive proof of the unimodality of the Gaussian polynomials in [0] are (1) the introduction of two auxiliary statistics, spread and degree, for the set of partitions whose Ferrers diagram fits in a given rectangle, (2) a Cartesian product decomposition of the set of partitions with given spread and degree, and (3) an inductive argument based on the Cartesian product decomposition.

O Section 4. Theorem 2 comes out of the next lemma when we sum over all ml ~ ... ~ m" for which ml + ... + m" = j. 51 LEMMA 5. Given 1 $ a $ k 0, M; + 1 and m} ~ '" = m} + ... + m;, the function ~ mk ~ mk+} = 0, Mo = is the generating function for partitions with at most a-II's, each block has size at most k, there are exactly mi - mi+1 blocks of size i, and if s( B) = i then weB) $ ni - min{i, a -I} - 2 E min{i,s(R)}. R Proof. It should be clear that peaks correspond to blocks, the weight of the peak to the weight of the bock, and the relative height of the peak to the size of the block.

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