Let n1,…,nk∈Z+ and n1+⋯+nk=n. The integer partition (n1,…,nk) of n is said to be realized if there is a latin square of order n with pairwise disjoint subsquares of order ni for each 1≤i≤k. In this paper, we construct latin squares realizing partitions of the form (as,bt); that is, partitions with s parts of size a and t parts of size b, where a<b. Heinrich (1982) showed that (1) if s≥3 and t≥3, then there is a latin square realizing (as,bt), (2) (as,b) is realized if and only if (s−1)a≥b, and (3) (a,bt) is realized if and only if t≥3. In this paper, we resolve the open cases. We show that (a2,bt) is realized if and only if t≥3 and (as,b2) is realized if and only if as≥b.
Journal of Combinatorial Designs – Wiley
Published: Jan 1, 2018
Keywords: ; ; ;
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