A generalization of Schönemann's theorem via a graph theoretic method

Recently, Grynkiewicz et al. [Israel J. Math. 193 (2013), 359–398], using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence a1x1+ · · ·+akxk ≡ b (mod n), where a1, . . . , ak, b, n (n ≥ 1) are arbitrary integers, has a solution 〈x1, . . . , xk〉 ∈ Z k n with all xi distinct modulo n. So, it would be an interesting problem to give an explicit formula for the number of such solutions. Quite surprisingly, this problem was first considered, in a special case, by Schönemann almost two centuries ago(!) but his result seems to have been forgotten. Schönemann [J. Reine Angew. Math. 1839 (1839), 231–243] proved an explicit formula for the number of such solutions when b = 0, n = p a prime, and ∑k i=1 ai ≡ 0 (mod p) but ∑ i∈I ai 6≡ 0 (mod p) for all I {1, . . . , k}. In this paper, we generalize Schönemann’s theorem using a result on the number of solutions of linear congruences due to D. N. Lehmer and also a result on graph enumeration recently obtained by Ardila et al. [Int. Math. Res. Not. 2015 (2015), 3830–3877]. This seems to be a rather uncommon method in the area; besides, our proof technique or its modifications may be useful for dealing with other cases of this problem (or even the general case) or other relevant problems.