Enumeration Problems for a Linear Congruence Equation

Let m ≥ 2 and r ≥ 1 be integers and let c ∈ Zm = {0, 1, . . . , m− 1}. In this paper, we give an upper bound and a lower bound for the number of unordered solutions x1, . . . , xn ∈ Zm of the congruence x1 + x2 + · · · + xr ≡ c mod m. Exact formulae are also given when m or r is prime. This solution number involves the Catalan number or generalized Catalan number in some special cases. Moreover, the enumeration problem has relationship with the restricted integer partition.