Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Integer partitions with fixed subsums. | Integer partitions with fixed subsums Yu. Yakubovich Department of Mathematics Utrecht University 80010 NL-3508 TA Utrecht The Netherlands Submitted Jan 17 2005 Accepted May 11 2005 Published May 16 2005 Mathematics Subject Classifications 05A17 Abstract Given two positive integers m n we consider the set of partitions A Al . A 0 . Al A2 . of n such that the sum of its parts over a fixed increasing subsequence aj is m Aai Aa2 m. We show that the number of such partitions does not depend on n if m is either constant and small relatively to n or depend on n but is close to its largest possible value n ma1 k for fixed k in the latter case some additional requirements on the sequence aj are needed . This number is equal to the number of so-called colored partitions of m respectively k . It is proved by constructing bijections between these objects. 1 Introduction In a recent paper 2 Canfield and his collaborators considered a set of partitions A A1 A2 . A1 A2 . of an integer n with a fixed sum of even parts . A2 A4 m . They in particular proved that the number of such partitions depends only on m for sufficiently large n namely for n 3m and equals to the number of colored partitions of m. These are partitions of m with each part having an additional attribute usually referred to as color which can take two values in this particular case. The number of such partitions f m is well known and the generating function for these numbersis X f mx n 1 by m 0 k 1 This research was supported by the NWO postdoctoral fellowship. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 N7 1 see . 1 . In other words as mentioned in 2 these numbers count ordered pairs of partitions A j such that I A 1 j m. Their proof is based on construction of a bijection between the set of partitions of n with the sum of even parts being equal to m and a pairs of partitions A j where I A IjI m and j has at most n 2m parts. We generalize this result in the following way. .
