Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Bijections and Congruences for Generalizations of Partition Identities of Euler and Guy. | Bijections and Congruences for Generalizations of Partition Identities of Euler and Guy James A. Sellers Andrew V. Sills and Gary L. Mullen Department of Mathematics The Pennsylvania State University University Park PA 16802 sellersj@ asills@ mullen@ Submitted Jul 6 2002 Accepted Jun 18 2004 Published Jun 25 2004. MR Subject Classifications 05A17 11P83 Abstract In 1958 Richard Guy proved that the number of partitions of n into odd parts greater than one equals the number of partitions of n into distinct parts with no powers of 2 allowed which is closely related to Euler s famous theorem that the number of partitions of n into odd parts equals the number of partitions of n into distinct parts. We consider extensions of Guy s result which naturally lead to a new algorithm for producing bijections between various equivalent partition ideals of order 1 as well as to two new infinite families of parity results which follow from Euler s Pentagonal Number Theorem and a well-known series-product identity of Jacobi. 1 Introduction A partition A of the integer n is a representation of n as a sum of positive integers wherein the order of the summands is considered irrelevant. Accordingly the summands can be rearranged in any order that seems convenient. Often in the literature the summands are placed in nonincreasing order but as we shall see other canonical representations of partitions are useful in various contexts. A summand in a partition is called a part of the partition. One of the most elegant partition identities known was discovered by Euler Current address Department of Mathematics Hill Center Busch Campus Rutgers University Piscataway NJ 08854 THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R43 1 Euler s Partition Identity. The number of partitions of n into odd parts equals the number of partitions of n into distinct parts. Proofs of this result abound 3 p. 5 9 p. 277 . In a brief note published in 1958 Richard Guy 8 gave a
