Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A Combinatorial Proof of Andrews’ Smallest Parts Partition Function. | A Combinatorial Proof of Andrews Smallest Parts Partition Function Kathy Qing Ji Center for Combinatorics LPMC-TJKLC Nankai University Tianjin 300071 . China ji@ Submitted Jan 14 2008 Accepted Mar 19 2008 Published Apr 10 2008 Mathematics Subject Classification 05A17 11P81 Abstract We give a combinatorial proof of Andrews smallest parts partition function with the aid of rooted partitions introduced by Chen and the author. 1 Introduction We adopt the common notation on partitions as used in 1 . A partition A of a positive integer n is a finite nonincreasing sequence of positive integers A A1 A2 Ar such that 52r 1 Ai n. Then Ai are called the parts of A. The number of parts of A is called the length of A denoted by l A . The weight of A is the sum of parts denoted by A . We let P n denote the set of partitions of n. Let spt n denote the number of smallest parts in all partitions of n and ns A denote the number of the smallest parts in A we then have spt n X ns A . Below is a list of the partitions of 4 with their corresponding number of smallest parts. We see that spt 4 10. A 2 P 4 ns A 4 1 3 1 1 2 2 2 2 1 1 2 1 1 1 1 4 THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N12 1 The rank of a partition A introduced by Dyson 6 is defined as the largest part minus the number of parts which is usually denoted by r A A1 l A . Let N m n denote the number of partitions of n with rank m. Atkin and Garvan 4 define the kth moment of the rank by Nk n XX mN m n . m i In 2 Andrews shows the following partition function on spt n analytically Theorem Andrews spt n np n 2 N2 n where p n is the number of partitions of n. At the end of the paper Andrews states that In addition the connection of N2 n 2 to the enumeration of 2-marked Durfee symbols in 3 suggests the fact that there are also serious problems concerning combinatorial mappings that should be investigated. In this paper we give a combinatorial proof of with the aid of rooted partitions .
