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Báo cáo toán học: "A Schr¨der Generalization of Haglund’s Statistic on o Catalan Paths"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A Schr¨der Generalization of Haglund’s Statistic on o Catalan Paths | A Schroder Generalization of Haglund s Statistic on Catalan Paths E. S. Egge Department of Mathematics Gettysburg College Gettysburg PA 17325 eggee@member.ams.org J. Haglund Department of Mathematics University of Pennsylvania Philadelphia PA 19104 jhaglund@math.upenn.edu K. Killpatrick Mathematics Department Pepperdine University Malibu CA 90263-4321 Kendra.Killpatrick@pepperdine.edu D. Kremeh Department of Mathematics Gettysburg College Gettysburg PA 17325 dkremer@gettysburg.edu Submitted May 6 2002 Accepted Apr 17 2003 Published Apr 23 2003 MR Subject Classifications 05A15 05E05 Abstract Garsia and Haiman J. Algebraic. Combin. 5 1996 191 244 conjectured that a certain sum Cn q t of rational functions in q t reduces to a polynomial in q t with nonnegative integral coefficients. Haglund later discovered Adv. Math. in press and with Garsia proved Proc. Nat. Acad. Sci. 98 2001 4313 4316 the refined conjecture Cn q t E qareatbounce. Here the sum is over all Catalan lattice paths and area and bounce have simple descriptions in terms of Partially supported by a Gettysburg College Professional Development Grant. Partially supported by a Gettysburg College Professional Development Grant. THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 R16 1 the path. In this article we give an extension of area bounce to Schroder lattice paths and introduce polynomials defined by summing qareatbounce over certain sets of Schroder paths. We derive recurrences and special values for these polynomials and conjecture they are symmetric in q t. We also describe a much stronger conjecture involving rational functions in q t and the V operator from the theory of Macdonald symmetric functions. 1 Introduction In the early 1990 s Garsia and Haiman introduced an important sum Cn q t of rational functions in q t which has since been shown to have interpretations in terms of algebraic geometry and representation theory. This rational function is defined explicitly in section 4 for now we wish to note