Divisor matrices and magic sequences

Abstract

Yuster (Arithmetic progressions with constant weight, Discrete Math. 224 (2000) 225–237) defines divisor matrices and uses them to derive results on “magic” sequences, i.e. finite sequences a 1, a 2,…, a n with the property that for a certain k all sums ∑ j=1 k a i j with i 1, i 2,…, i k an arithmetic subsequence of 1,2,…, n, are equal. An important condition is the (conjectured) non-singularity of the elementary divisor matrices A k , that could only be proved for k with at most two prime divisors. We present a proof for general k, thereby generalizing the results in Yuster [1] (Arithmetic progressions with constant weight, Discrete Math., to appear.). Our exploration of A k also leads to new proofs, and enables us to add other results, in particular we give the dimension of the space of k-magic sequences of length n for every k and n and over every field.