关键词: 正整数的有序分拆, n-color 1-2-3有序分拆, 回文的n-color 1-2-3有序分拆, 生成函数, 显式公式, 递推公式, 组合证明

Abstract: An n-color 1-2-3 composition of positive integers is defined as an n-color composition with only parts of size 1, 2 or 3. An n-color 1-2-3 palindromic composition is an n-color 1-2-3 composition that reads the same forward as backward. Here the generating function, explicit formulas and recurrence relations for the number of n-color 1-2-3 compositions and the n-color 1-2-3 palindromic compositions of positive integers were obtained. In addition,  a relation between the number of  1-2-3 compositions of a positive integer and the number of compositions of a positive integer with parts omitting all multiples of size 3 was given. Furthermore, the generalized relation was obtained.

Key words: compositions of positive integers, n-color 1-2-3 composition, n-color 1-2-3 palindromic composition, generating function, explicit formula, recurrence relation, combinatorial proof