Integer partitions of zero and compositional structures in finite groups

This paper addresses the enumeration of two types of zero-sum sequences. The first type is defined by constraints on both the number of terms and the values they may assume; the second type is constrained by the number of terms and the total absolute value of the sequence. For both cases, exact enumeration formulas are derived in terms of restricted partition functions, and asymptotic estimates are obtained. In the final section, an algebraic analogue of restricted compositions is introduced, and its enumerative properties are analyzed.