tailieunhanh - Báo cáo toán học: "Partition statistics for cubic partition pairs"

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: Partition statistics for cubic partition pairs. | Partition statistics for cubic partition pairs Byungchan Kim Department of Liberal Arts Seoul National University of Science and Technology Seoul Korea bkim4@ Submitted Mar 17 2011 Accepted May 31 2011 Published Jun 14 2011 Mathematics Subject Classification 05A17 11P83 Abstract In this brief note we give two partition statistics which explain the following partition congruences b 5n 4 0 mod 5 b 7n a 0 mod 7 if a 2 3 4 or 6. Here b n is the number of 4-color partitions of n with colors r y o and b subject to the restriction that the colors o and b appear only in even parts. 1 Introduction In a series of papers 3 4 5 . Chan studied congruence properties of a certain kind of partition function a n which arises from Ramanujan s cubic continued fraction. This partition function a n is defined by fi 1 n 0a n q q qMq2 q2o Here and in the sequel we will use the following standard q-series notation tt a qK n 1 - aqn-1 q 1 n 1 Since a partition congruence for a n is deduced from the equation for Ramanujan s cubic continued fraction ZA q1 3 q q2 q2 q4 v q l 1 q 1 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P128 1 see 3 for the details. a n is known as the number of cubic partitions. After Chan s works many analogous partition functions have been studied. In particular H. Zhao and Z. Zhong 7 investigated congruences for the partition function ro 1 1 q . Here b n counts the number of partition pairs Al A2 where Al and A2 are cubic partition such that the sum of parts in A1 and A2 equals to n. In this sense we will call b n the number of cubic partition pairs. We can interpret b n as the number of 4-color partitions of n with colors r y o and b subject to the restriction that the colors o and b appear only in even parts. For example there are 7 such partitions as follows 2 2y 2o 2b 1r 1r. 1r 1y 1y 1y. Once congruence properties of a certain type of partition function are known it is natural to seek a partition statistic to give a combinatorial explanation