2021/07/19 by Fiorenza Morini, Morini, Fiorenza, Marco Antonio Pellegrini +1 · 1 citation
Agricultural and Biological Sciences · #05B20 #Botanical Research and Chemistry #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2107.08857
openalex publication_date 2021/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let m,n,s,k be four integers such that 3≤ s ≤ n, 3≤ k≤ m and ms=nk. Set d=gcd(s,k). In this paper we show how one can construct a Heffter array H(m,n;s,k) starting from a square Heffter array H(nk/d;d) whose elements belong to d consecutive diagonals. As an example of application of this method, we prove that there exists an integer H(m,n;s,k) in each of the following cases: (i) d≡ 0 \pmod 4; (ii) 5≤ d≡ 1 \pmod 4 and n k≡ 3\pmod 4; (iii) d≡ 2 \pmod 4 and nk≡ 0 \pmod 4; (iv) d≡ 3 \pmod 4 and n k≡ 0,3\pmod 4. The same method can be applied also for signed magic arrays SMA(m,n;s,k) and for magic rectangles MR(m,n;s,k). In fact, we prove that there exists an SMA(m,n;s,k) when d≥ 2, and there exists an MR(m,n;s,k) when either d≥ 2 is even or d≥ 3 and nk are odd. We also provide constructions of integer Heffter arrays and signed magic arrays when k is odd and s≡ 0 \pmod 4.