自然数集的分拆及其表示函数

Partitions of Natural Numbers and Their Representation Functions

令N表示全体非负整数的集合.对给定的集合A C N及n∈N,令R_1(A,n)表示方程n=a+a',a,a'∈A的解的个数.令R_2(A,n)和R_3(A,n)分别表示方程n=a+a',a,a'∈A在条件a<a'和a≤a'下解的个数.一个有趣的问题是:给定i∈{1,2,3},确定所有非负整数集合对(A;B),使其表示函数R_i(A,n)及R_i(B,n)最终相等.文章讨论了相关问题.

Let N be the set of all nonnegative integers. For A C N and n ∈ N, let R1 （A, n） be the number of solutions of n = a ＋ a＇, a, a＇ ∈ A, and let R2（A, n） and R3（A, n） be the numbers of solutions with the additional restrictions a 〈 a＇ and a ≤ a＇, respectively. One of the interesting problems in connection with representation functions is： For any given i ∈ （1, 2, 3）, describe all pairs of sets of nonnegative integers （A; B） whose representation functions Ri（A, n） and Pi（B, n） eventually coincide. Some related problems are discussed.