Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: On a Partition Function of Richard Stanley. | On a Partition Function of Richard Stanley George E. Andrews Department of Mathematics The Pennsylvania State University University Park PA 16802 andrews@ Submitted Sep 5 2003 Accepted Nov 19 2003 Published Jun 3 2004 MR Subject Classifications 05A17 In honor of my friend Richard Stanley Abstract In this paper we examine partitions n classified according to the number r n of odd parts in n and s n the number of odd parts in n the conjugate of n. The generating function for such partitions is obtained when the parts of n are all 5 N. From this a variety of corollaries follow including a Ramanujan type congruence for Stanley s partition function t n . 1 Introduction Let n denote a partition of some integer and n its conjugate. For definitions of these concepts see 1 . Let O n denote the number of odd parts of n. For example if n is6 5 4 2 2 1 then the Ferrers graph of n is Reading columns we see that n is6 5 3 3 2 1. Hence O n 2 and O n 4. Richard Stanley 4 and 5 has shown that if t n denotes the number of partitions n of n for which O n O n mod 4 then Partially supported by National Science Foundation Grant DMS-0200047 THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2004 R1 1 X 1 t n 2 p n f n 1 where p n is the total number of partitions of n 1 p. 1 and .f n r n n 0 i 1 1 q2i-1 1 - q4i 1 q4i-2 2 2 Note that t n is Stanley s partition function referred to in the title of this paper. Stanley s result for t n is related nicely to a general study of sign-balanced labeled posets 5 . In this paper we shall restrict our attention to SN n r s the number of partition n of n where each part of n is 5 N O n r O n s. In Section 2 we shall prove our main result Theorem 1. 52 S2N n r s qnzrys n r s 0 P-N 2 j 0 N q4 -zyq q4 j -zy 1 q q4 N-j yq 2N 2j q4 q4 N z2q2 q4 N 3 and 52 S2N 1 n r s qnzrys n r s 0 N q4 -zyq q4 j i -zy 1 q q4 N-j yq 2N 2j q4 q4 N z2q2 q4 N 1 N j 0 4 where q 1-qN 1-qN-1 . 1-qN-j 1 for 0 5 j 5 N if j 0 or j N 5 N _j 1-qj 1-ại-1 . 1-ợ 0 and A q M 1
