Order-restricted linear congruences

Abstract

We call the congruence a1x1+⋯+akxk≡b(modn) an order-restricted linear congruence if x1≥⋯≥xk. What can we say about the number of solutions of these congruences? In this paper we consider the special case of ai=1(1≤i≤k), and using a result from the theory of partitions and also properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, give an explicit formula for the number of solutions of such congruences. This generalizes the result of Riordan (1962) which gives an explicit formula for the special case of ai=1(1≤i≤k), k=n, b=0.