tailieunhanh - Báo cáo toán học: "The polynomial part of a restricted partition function related to the Frobenius problem"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: The polynomial part of a restricted partition function related to the Frobenius problem. | The polynomial part of a restricted partition function related to the Frobenius problem Matthias Beck Ira M. Gessel Department of Mathematical Sciences State University of New York Binghamton NY 13902-6000 USA matthias@ Department of Mathematics Brandeis University Waltham MA 02454-9110 USA gessel@ Takao Komatsu Faculty of Education Mie University Mie 514-8507 Japan komatsu@ Submitted May 29 2001 Accepted September 4 2001 MR Subject Classifications Primary 05A15 Secondary 11P81 05A17 Abstract Given a set of positive integers A pl . an we study the number pA t of nonnegative integer solutions m1 . mn to 52n 1 mjaj t. We derive an explicit formula for the polynomial part of PA. Let A a1 . an be a set of positive integers with gcd a1 . an 1. The classical Frobenius problem asks for the largest integer t the Frobenius number such that m1a1 mnan t has no solution in nonnegative integers m1 . mn. For n 2 the Frobenius number is a1 1 a2 1 1 as is well known but the problem is extremely difficult for n 2. For surveys of the Frobenius problem see R Se . One approach BDR I K SO is to study the restricted partition function pA t the number of nonnegative integer solutions m1 . mn to 52n 1 mjaj t where t is a nonnegative integer. The Frobenius number is the largest integral zero of pA t . Note that in contrast to the Frobenius problem in the definition of Pa we do not require a1 . an to be relatively prime. In the following a1 . an are arbitrary positive integers. Research partially supported by NSF grant DMS-9972648. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 N7 1 It is clear that pA t is the coefficient of z in the generating function G z ----1 --------- T 1 zai 1 zan If we expand G z by partial fractions we see that rpA t can be written in the form X Paw. X where the sum is over all complex numbers A such that Aai 1 for some i and PA X t is a polynomial in t. The aim of this paper is to give an explicit formula for PA i t .

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