On the number of solutions to linear Diophantine equation and Frobenius problem

Issues concerning the solvability and number of solutions to linear Diophantine equations are considered. Along with the general case, combinatorial characteristics of the number of solutions and the mean number of solutions to equations of a special case are studied. One type of equation represents partitions of a natural number into natural additive components. Another type consists of linear equations in two unknowns that are usually studied in relation to the Frobenius problem. The focus is on three aspects. The first is the solvability and number of solutions of the Diophantine equation when the problem is parameterized with respect to the right-hand side. Formulas and bounds for finding this number both in the general case and in some particular cases are obtained. The second aspect is devoted to the partition problem. The third aspect concerns the Frobenius problem.