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Báo cáo toán học: "From Recursions to Asymptotics"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: From Recursions to Asymptotics. | From Recursions to Asymptotics On Szekeres Formula for the Number of Partitions E. Rodney Canfield Department of Computer Science University of Georgia Athens gA 30602 UsA erc@cs.uga.edu For Herb Wilf on his 65-th Birthday Submitted August 1 1996 Accepted November 21 1996 Abstract. We give a new proof of Szekeres formula for P n k the number of partitions of the integer n having k or fewer positive parts. Our proof is based on the recursion satisfied by P n k and Taylor s formula. We make no use of the Cauchy integral formula or any complex variables. The derivation is presented as a step-by-step procedure to facilitate its application in other situations. As corollaries we obtain the main term of the Hardy-Ramanujan formulas for p n the number of unrestricted partitions of n and for q n the number of partitions of n into distinct parts. AMS-MOS Subject Classification 1990 . Primary 05A17 Secondary 05A20 05A16 11P81 THE ELECTRONIC JOURNAL OF COMBINATORICS 4 no. 2 1997 R6 2 1 Introduction. A partition of an integer n into k parts is a solution to the system n X1 x2 Xk X1 x2 Xk 0. Let P n k be the number of partitions of n into k or fewer parts. We will prove the following. Theorem. Szekeres Let e 0 be given. Then uniformly for k n1 6 P n k f u exp I n1 2g u O n 1 6 eVl. n Here u k n1 2 and the functions f u g u are v f u 93 L Í1 - e v - 2u2e v 1 2 1-1 23 2 nu 2v z_ _ g u -ulog 1 -e 1-2 u where v v u is determined implicitly by u2 v2 Ị Ị 1 dt. 1.3 Remarks. The estimate can be made uniform for the entire range k 1 by adding 1 k to the big-oh term. The last equation uniquely determines v because the right hand side is an increasing function of v . Szekeres presents his results in two papers 12 13 using substantially different approaches for two distinct though slightly overlapping ranges of k . The papers are remarkable both for the depth of the analysis contained in them and for the precision of their results. Indeed Szekeres is the only known proof that p n k is .