Báo cáo toán học: "On Stanley’s Partition Function"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài:On Stanley’s Partition Function. | On Stanley s Partition Function William Y. C. Chen1 Kathy Q. Ji2 and Albert J. W. Zhu3 Center for Combinatorics LPMC-TJKLC Nankai University Tianjin 300071 . China 1chen@ 2ji@ 3zjw@ Submitted Jun 12 2010 Accepted Aug 19 2010 Published Sep 1 2010 Mathematics Subject Classification 05A17 Abstract Stanley defined a partition function t n as the number of partitions A of n such that the number of odd parts of A is congruent to the number of odd parts of the conjugate partition A modulo 4. We show that t n equals the number of partitions of n with an even number of hooks of even length. We derive a closed-form formula for the generating function for the numbers p n t n . As a consequence we see that t n has the same parity as the ordinary partition function p n . A simple combinatorial explanation of this fact is also provided. 1 Introduction This note is concerned with the partition function t n introduced by Stanley 8 9 . We shall give a combinatorial interpretation of t n in terms of hook lengths and shall prove that t n and the partition function p n have the same parity. Moreover we compute the generating function for p n t n . We shall adopt the common notation on partitions in Andrews 1 or Andrews and Eriksson 3 . A partition A A1 A2 A3 . Ar of a nonnegative integer n is a nonincreasing sequence of nonnegative integers such that the sum of the components Ai equals n. A part is meant to be a positive component and the number of parts of A is called the length denoted l A . The conjugate partition of A is defined by A A1 A2 . At 1 2 t where Ai 1 i t t l A is the number of parts in A1 A2 . Ar which are greater than or equal to i. The number of odd parts in A A1 A2 . Ar is denoted by O A . For q 1 the q-shifted factorial is defined by a q n 1 a 1 aq 1 aqn-1 n 1 and a qW 1 a 1 aq 1 aq2 THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N31 1 see Gasper and Rahman 5 . Stanley 8 9 introduced the partition function t n as the .