Enumeration problems for a linear congruence equation

Abstract

Let m > 2 and r > 1 be integers and let c e 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 e 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.