RESTRICTED PARTITION FUNCTIONS AND ADDITIVE COMPLEMENTS

Abstract Let N ${\mathbb N}$ double struck upper N be the set of positive integers. For subsets A , M ⊆ N $\mathcal {A},\mathcal {M}\subseteq {\mathbb N}$ script upper A comma script upper M subset of or equal to double struck upper N and n ∈ N $n\in {\mathbb N}$ n element of double struck upper N , let p ( n , A , M ) $p(n,\mathcal {A},\mathcal {M})$ p left parenthesis n comma script upper A comma script upper M right parenthesis denote the number of representations of n in the form n = ∑ a ∈ A m a a $n=\sum _{a\in \mathcal {A}}m_a a$ n equals sigma summation Underscript a element of script upper A Endscripts m Subscript a Baseline a , where m a ∈ M ∪ { 0 } $m_a\in \mathcal {M}\cup \{0\}$ m Subscript a Baseline element of script upper M union StartSet 0 EndSet for all