Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: The umbral transfer-matrix method IV. Counting self-avoiding polygons and walks. | The umbral transfer-matrix method IV. Counting self-avoiding polygons and walks Doron ZEILBERGER 1 zeilberg@ Submitted March 13 2001 Accepted July 26 2001. MR Subject Classifications 05A To think in a computerized way is an important matter . to go with it to the end of the limit of possibilities and there to develop new unpredictable ones. David Avidan Free translation of an excerpt from the Hebrew original of Avidan s poem lakhshov betsura memukhshevet To Think In a Computerized Way . Abstract This is the fourth installment of the five-part saga on the Umbral Transfer-Matrix method based on Gian-Carlo Rota s seminal notion of the umbra. In this article we describe the Maple packages USAP USAW and MAYLIS. USAP automatically constructs for any specific r an Umbral Scheme for enumerating according to perimeter the number of self-avoiding polygons with 2r horizontal edges per vertical cross-section. The much more complicated USAW does the analogous thing for self-avoiding walks. Such Umbral Schemes enable counting these classes of self-avoiding polygons and walks in polynomial time as opposed to the exponential time that is required by naive counting. Finally MAYLIS is targeted to the special case of enumerating classes of saps with at most two horizontal edges per vertical cross-section equivalently column-convex polyominoes by perimeter and related classes. In this computationally trivial case we can actually automatically solve the equations that were automatically generated by USAP. As an example we give the first fully computer-generated proof of the celebrated Delest-Viennot result that the number of convex polyominoes with perimeter 2n 8 equals 2n 11 4n 4 2n 1 n 2. The Third and Ultimately Most EFFICIENT Way of Using MAPLE The great combinatorial enumerator Mireille Bousquet-Mélou in her fascinating Rapport B p. 20 writes about the two ways she uses Maple. The first more traditional one is en aval downstream . after the research which .
